POSE DETECTION AND CONTROL OF UNMANNED UNDERWATER VEHICLES (UUVs) UTILIZING AN OPTICAL DETECTOR ARRAY

ABSTRACT

Optical detectors and methods of optical detection for unmanned underwater vehicles (UUVs) are disclosed. The disclosed optical detectors and may be used to dynamically position UUVs in both static-dynamic systems (e.g., a fixed light source as a guiding beacon and a UUV) and dynamic-dynamic systems (e.g., a moving light source mounted on the crest of a leader UUV and a follower UUV).

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 62/145,077, filed on Apr. 9, 2015, which is herebyincorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

Unmanned underwater vehicles (UUVs) play a major role in deep oceanicapplications, such as underwater pipeline and cable inspection,bathymetry exploration as well as in military applications such as minedetection, harbor monitoring and anti-submarine warfare. Theseapplications mostly take place in deep sea environment and include heavyduty tasks that may take long time periods and therefore, are notsuitable to be performed by divers. Some underwater operations (e.g.,surveying a large area or an area with a complex seafloor bathymetry)require more than one Unmanned Underwater Vehicle (UUV) for efficienttask completion. In these cases, the deployment of multiple UUVs information can perform such tasks and reduce the operational time andcosts.

SUMMARY OF THE INVENTION

In laboratory conditions at University of New Hampshire (UNH) Jere A.Chase Ocean Engineering facilities, it has been demonstrated that anoptical detector array is capable of discriminating pose in threetranslational axes, x, y and z. In addition, the optical detector arrayis capable of generating velocity signal as feedback to the UnmannedUnderwater Vehicle control system. The accuracy of the pose estimationswith respect to a light source as a guiding beacon is within 0.5 m inx-axis and 0.2 m for y and z-axes. The velocity estimations in x-axisare within 0.14 m/s.

In order to predict the pose estimation performance in different waterconditions such as in Portsmouth Harbor, N.H., uncertainty analysis hasbeen conducted through Monte Carlo simulations. The simulations resultssuggest that under the calibration conditions conducted at UNH OceanEngineering facilities, the pose estimations performance in predictedPortsmouth Harbor water conditions decrease by 1 m in x-axis, 0.05 m fory-axis and 0.2 m for z-axis.

As part of the research for development of a leader-follower formationbetween unmanned underwater vehicles (UUVs), this study presentsutilization of optical feedback for UUV navigation by developing anoptical detector array. Capabilities of pose detection and control in astatic-dynamic system (e.g., a UUV navigation into a docking station)and a dynamic-dynamic system (UUV-UUV leader-follower) are investigated.In both systems, a single light source is utilized as a guiding beaconfor the UUV and an optical array consisting of photodiodes is used toreceive the light field emitted from the light source.

For UUV navigation applications, accurate pose estimation is important.In order to evaluate the feasibility of underwater distance detection,experimental work is conducted. Based on the experiments, the range ofoperations between two platforms, i.e., light source and opticaldetector, the optimum spectral range that allowed maximum lighttransmission is calculated. Based on the light attenuation inunderwater, dimensions of an optical detector array are determined. Theboundary conditions for the pose detection algorithms and the errorsources in the experiments are identified.

As a test-bed to determine optical array dimensions and size, asimulator, i.e., numerical software, is developed. In the simulator,planar and curved array geometries of varying number of elements areanalytically compared and evaluated. Results show that the curvedoptical detector array is able to distinguish 5-DOF motion (translationin x, y, z and pitch and yaw rotations) with respect to the single lightsource. The positional changes of 0.2 m and rotational changes of 10 owithin 4 m-8 m range in x-axis can be detected. Analytical posedetection and control algorithms are developed for both static-dynamic(UUV-docking station) and dynamic-dynamic (UUV-UUV) systems for dynamicpositioning applications. Three different image processing algorithms,i.e. phase correlation and log-polar transform, Spectral Angle Mapper(SAM), and image moment invariants, are evaluated for pose detection ofthe UUVs. The efficacy of Proportional-Integral-Derivative (PID) andSliding Mode Controller) SMC is evaluated for static-dynamic systems.The algorithms are developed for curved optical detectors of size 21×21and 5×5 for varying single DOF static-dynamic system controls andmultiple initial condition static-dynamic control as well asdynamic-dynamic control. Results show that a 5×5 detector array with theimplementation of SMC is sufficient for UUV dynamic positioningapplications.

The capabilities of an optical detector array to determine the pose of aUUV in 3-DOF (x, y and z-axes) are experimentally tested. Anexperimental platform consisting of a 5×5 photodiode array mounted on ahemispherical surface is used to sample the light field emitted from asingle light source. Pose geometry calibrations are conducted bycollecting images taken at 125 positions in 3-D space. Pose detectionalgorithms are developed to detect pose for steady-state and dynamiccases. Monte Carlo simulations are conducted to assess the poseestimation uncertainty under varying environmental and hardwareconditions such as water turbidity, temperature variations in water andelectronics noise. Experimental results demonstrate that x, y and z-axispose estimations are accurate within 0.3 m, 0.1 m and 0.1 mrespectively. Monte Carlo simulation results show that the poseuncertainties (95%) associated with x, y and z-axes are 1.5 m, 1.3 m and1 m, respectively.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features, and advantages of theinvention will be apparent from the following description of particularembodiments of the invention, as illustrated in the accompanyingdrawings in which like reference characters refer to the same partsthroughout the different views. The drawings are not necessarily toscale, emphasis instead being placed upon illustrating the principles ofthe invention.

FIG. 1A shows a body-fixed reference frame and an earth fixed referenceframe.

FIG. 1B shows the earth-fixed non-rotating reference frame and thebody-fixed rotating reference frame X_(o)Y_(o)Z_(o).

FIG. 1C shows two dimensional added mass coefficients used in striptheory.

FIG. 1D shows a UUV Control Block diagram with the output obtained fromoptical feedback array. Controller regulates the UUV motion based on thefeedback obtained from the optical detector array and changes the courseof the UUV by sending commands to the thrusters.

FIG. 2A shows an experimental schematic of the UNH tow tank.

FIG. 2B shows an experimental Setup for translational 3-D underwaterexperiments. (Left image) detector unit that includes a submerged fiberoptic cable with a collimator that was connected to the spectrometer.(Right image) transmitting unit mounted to the wall of the tank.

FIG. 2C is a percent attenuation graph. This graph shows the lightpercent attenuation per meter. It is seen that the spectral rangebetween 500-550 nm undergoes the least attenuation at any givendistance.

FIG. 2D shows an intensity vs. distance plot. The intensity readings arecollected between 500-550 nm and averaged. In this plot, theexperimental values are compared with the theoretical. Blue diamondsrepresent the experimental data and the green triangles represent thetheoretical calculations from taking the inverse square and Beer-Lambertlaws. The readings are normalized. The measurement at 4.5 m was used asthe reference measurement to normalize the intensity.

FIG. 2E is a plot of the cross-sectional beam pattern. The measurementswere collected from 0 to 1.0 m at x-axis and at 4.5 m at theillumination axis for 50 W light source. The measurements between500-550 nm are average.

FIG. 3A is a schematic illustration of array designs used in thesimulator: (a) Planar array and (b) Curved array.

FIG. 3B shows key image parameters and intensity profiles for a planararray detector unit with hardware and environmental background noise:(top left) Output image from the simulator, (top right) Horizontalprofile, (bottom left) Vertical profile, (bottom right) Input valuesused to generate output image and key parameters describing outputimage.

FIG. 3C shows key image parameters and intensity profiles for a curvedarray detector unit with hardware and environmental background noise:(top left) Output image from the simulator, (top right) Horizontalprofile, (bottom left) Vertical profile, (bottom right) Input valuesused to generate output image and key parameters describing outputimage.

FIG. 3D illustrates comparative resemblance results (SAM angles) for21×21 element curved and planar array (at x=4 m) as a function of: (a)lateral translation, (b) yaw rotation.

FIG. 3E illustrates comparative resemblance results (i.e., SAM angle)with respect to varying array sizes (incorporating environmental andbackground noise): (a) SAM angle with respect to lateral motion (b) SAMangle with respect to angular rotation.

FIG. 3F shows comparative resemblance results (i.e., SAM angle) withrespect to operational distance (incorporating environmental andbackground noise): (a-c) lateral shift, (d-f) yaw rotation—(a,d) 3×3array (b,e) 5×5 array (c,f) 101×101 array with spacing of 0.2 m, 0.1 mand 0.004 m, respectively.

FIG. 3G shows a comparison of Experimental and Simulation results (a) 4m (b) 5 m (c) 6 m (d) 7 m (e) 8 m.

FIG. 4A illustrates transformation of an image from Cartesian space topolar space. Cartesian space (left). Polar space (right).

FIG. 4B illustrates independent DOF SMC results for a curved 21×21array. (a) x-axis control (b) y-axis control (c) z-axis control (d) yawcontrol (e) pitch control.

FIG. 4C shows independent DOF control results with SMC for a curved 5×5array. (a) x-axis control (b) y-axis control (c) z-axis control (d) yawcontrol (e) pitch control.

FIG. 4D shows a PID x-axis control for a 5×5 array.

FIG. 4E illustrates a UUV docking case study using SMC for a 5×5 array.The UUV with four initial non-zero pose errors is commanded to positionitself with respect to a fixed light source. (a) x-axis control (b)y-axis control c) yaw control d) z-axis control.

FIG. 4F illustrates a UUV docking case study using SMC for a 5×5 arraywith a current of −0.03 m/s in x-axis. (a) x-axis control (b) y-axiscontrol c) yaw control d) z-axis control.

FIG. 4G shows the leader-follower case study in a dynamic-dynamic systemwith SMC for a 5×5 array. (a) x-axis control (b) y-axis control and (c)z-axis control.

FIG. 5A shows a funnel type docking station.

FIG. 5B shows pole docking mechanism architectures.

FIG. 5C shows the optical detector array used in the experiments. Thephotodiodes are facing different angles for an increased field-of-view.They are placed on an ABS hemisphere surface for precise hole locations.The acrylic hemisphere is used for waterproofing. Left—Top view.Right—Side view.

FIG. 5D shows the Wave and Tow tank at the UNH Ocean Engineeringfacilities. The Tow Tank is cable driven and computer controlled with 1mm precision along x-axis. Left—detector array mounted on the dynamicplatform on the Tow-Tank. Right—400 W light beacon used as a mock-updocking station.

FIG. 5E shows a Photodiode Calibration Procedure. 25 photodiodes weretested at a time in order to observe their voltage range under sameconditions.

FIG. 5F shows a diagram for temperature calibration and an experimentalsetup for photodiode response to temperature changes. The diagram fortemperature calibration is shown at the top of the Figure. Athermocouple was placed inside the waterproof housing for temperaturemonitoring of the photodiode. The experimental setup for photodioderesponse to the temperature changes is shown at the bottom of theFigures.

FIG. 5G is a Monte Carlo flow diagram for the pose statistics.Pre-determined model uncertainty parameters were integrated into thehardware and environment model to estimate the total uncertaintypropagation in the pose detection algorithms.

FIG. 5H shows Experimental Case 1: Top left: Reference position, rawx-axis pose estimate, corrected x-axis pose estimates, and the movingaverage window result. Top right: Velocity reference, raw velocityestimates and moving average window of size 10 applied to the rawvelocity estimates. Bottom left: y-axis pose estimate and the appliedmoving average window of size 10 during the motion. Bottom right: z-axispose estimate and the applied moving average of size 10 during themotion.

FIG. 5I shows Experimental Case 2: Top left: Reference position, rawx-axis pose estimate, corrected x-axis pose estimates, and the movingaverage window of size 10 result. Top right: Velocity reference, rawvelocity estimates and moving average window of size 10 applied to theraw velocity estimates. Bottom left: y-axis pose estimate and theapplied moving average window of size 10 during the motion. Bottomright: z-axis pose estimate and the applied moving average of size 10during the motion.

FIG. 5J illustrates an experimental pose with the Monte Carlo generatedCI bounds (95%). The standard deviation of the hardware noise is set to1% intensity of the photodiode with the maximum intensity.

FIG. 5K shows Monte Carlo Simulation results with 95% CI bounds when theconceptual UUV navigates a zig-zag trajectory in-plane. The standarddeviation of hardware noise is set to 0.5% intensity of the photodiodewith the maximum intensity. Top-left: Nominal x-axis pose estimations.Top-right: y-axis pose estimation. Middle-left: z-axis pose estimation.Middle-right: Yaw pose estimation. Bottom: UUV reference navigation inthe x-y plane and the nominal estimation.

FIG. 5L is a closer look at the cross-talk between yaw and y-axiscross-talk when there is only yaw motion. At x-axis increments of 0.1 m,the conceptual UUV was rotated from −15 o to 15 o at 3 o increments.Monte Carlo simulation was conducted with sample number of 2000 andhardware noise level of 0.5% of the photodiode with the maximumintensity.

FIG. 6A illustrates a Photodiode-PC communication general diagram.

FIG. 6B shows an Arduino sketch. This will be for one Arduino. For theother Arduino change the 13 to 12 and other variables accordingly.

FIG. 6C shows a BB-XM setup.

FIG. 6D shows a Minicom login screen.

FIG. 6E shows a serial to USB port check on PC.

FIG. 6F shows an arm login and password screen.

FIG. 6G shows Arduino Device names verified in the BB-XM.

FIG. 6H shows a program that reads data from two Arduinos and passes itto the pc. (readPD.py)

FIG. 6I shows a program that reads the serial output of BB-XM and savesit to a file (getPD.py)

FIG. 7A shows beam pattern images at x=4.5 m

FIG. 7B shows beam pattern images at x=5.5 m

FIG. 7C shows beam pattern images at x=6.5 m

FIG. 7D shows beam pattern images at x=7.5 m

FIG. 7E shows beam pattern images at x=7.5 m

DETAILED DESCRIPTION

A requirement for a group of UUVs to move in a controlled formation isan underwater communication link between the UUVs. In addition to UUVoperation in formation, underwater communication links can also be usedfor UUV docking or data transfer from an operating UUV to a data storageplatform. The two latter applications allow UUVs to operate with longerperiods underwater without the need for excessive emerging/submerging.This study presents the development of an optical feedback interface andcontrol system for two types of UUV applications: 1) Static-Dynamicsystem (e.g., a UUV and a data transfer/storage platform such as adocking station) and 2) Dynamic-Dynamic system (i.e., formation controlof at least two UUVs). A requirement for a fleet of UUVs to move in acontrolled formation is a reliable underwater communication link betweenall UUVs and between the UUV to a docking station.

There is a variety of possible formation architectures for controlledformation of unmanned vehicles. Most of these architectures requirespecialized on-board hardware to enable communication between thevehicles in formation. For coordinated formation control of unmannedvehicles, a variety of formation architectures and strategies have beendeveloped. The main strategies include:

Virtual structure approach—In this approach, the whole fleet is treatedas a single rigid structure. The main advantage of this approach is thata highly precise formation can be maintained. However, its disadvantageis that the position and orientation from the agents' states requireshigh computational complexity.

Behavior based methods—Several behaviors for each robot are employed andfinal control action is obtained from the weighting of each behavior.However, the stability of the system is not guaranteed because there isnot enough modeling information for the subsystems and the environment.

Leader-follower—This method employs one vehicle as the leader thatguides the other vehicles in the formation (the followers). Based onone-way communication transmitted from the leader, the followersposition themselves relative to the leader position and orientation. Theleader-follower method is considered less complex than the otherapproaches as it requires no feedback from the followers to the leader.The disadvantage of this method is that if there is an error in theleader's trajectory, the followers deviate from their trajectory as welland the error accumulates.

Artificial potentials—In artificial potentials an interaction controlforce is defined between the vehicles. The artificial potential use thisforce to enforce a desired inter-vehicle spacing. In this method, thereis no leader vehicle assigned in the fleet. This eliminates the singlepoint failures and adds robustness to the system. However, theassumption is that each node is equipped with a sensor allowing it todetermine the range and forces between each node, which increase thenumber of hardware and complexity in the system.

Graph theory—Graph theory allows flexibility in changing the groupformation during the operation. However, this approach needs a list ofall possible transition geometries that is expected to occur in therobots that are in the formation. In addition, a good plan of action isneeded when faced with environmental and sensor constraints.

The formation control approach used in this study, more specifically inthe dynamic-dynamic system, is the leader-follower strategy because ofthe simplicity in its implementation. In an underwater environment, thecommunication signals commonly used in aerial and terrestrial vehicles(e.g., GPS and radio signals) are significantly attenuated and thuscannot be used.

Most studies on inter-communication between UUVs have concentrated onacoustic communication that performs well over long distances. Acousticcommunication types used in underwater operations consist of LongBaseline (LBL), Short Baseline (SBL) and Ultra-Short Baseline (USBL)systems. In LBL, multiple acoustic transponders are placed on theseafloor and provide high accuracy navigation for underwater tasks thatrequire precision. LBL systems are used in leader-follower formationflying systems SBL systems are mainly used for tracking of theunderwater vehicles and divers. Unlike LBL systems, SBL transponders arenot placed on the seafloor. Multiple SBL transponders are placed inwater from the sides of the ship and one transponder is placed on thetarget to be tracked. SBL systems are used in UUV to docking stationcommunication. USBL systems which offer fixed precision consist of twotransponders, one is lowered to the sea on the ship and the other one isplaced on the target of interest. In addition, USBL systems have foundapplication in docking systems as well. However, the necessary hardwarefor acoustics communication is costly and requires payloadconsiderations in the UUV platform design. In areas with large trafficvolume, such as harbor and recreational fishing areas, the marineenvironment can become acoustically noisy that can reduce theperformance of the acoustic communication and may not allow UUVoperations such as docking.

A cost-effective alternative is optical detection that either usesexisting hardware (e.g., light sources as beacons) or additionalhardware, i.e. low cost, commercially available off the shelf (COTS)photo detectors, etc. In astronautical and aeronautical applications,optical communications are used for navigation, docking and datatransfer. For example, free space optical communication is used inrendezvous radar antenna systems. In both cases of interspacecraftrendezvous and docking, a continuous-wave laser is transmitted from thepursuer spacecraft to a target spacecraft or to aid in the dockingprocess. The challenge to conduct underwater optical communication isthat light is significantly more scattered and absorbed in water than itis in air. As a result, the effective communication range is, however,shorter than that of acoustic communication. Optical communication fordata transfer in underwater was demonstrated at range of 30 m for clearwater conditions. In addition to relatively shorter range of operation,the optical properties of water (e.g., diffuse attenuation coefficientand scattering) constantly change and affect communication reliability.

In some examples, detector arrays consisting of individual opticaldetector elements are used for pose detection between UUVs. Manypossible geometric shapes for optical detector arrays exist. In variousexamples, two array designs are presented: planar and curved. Eachdesign has its own benefits. A planar-array design can maximize thesignal-to-noise ratio between all its elements, while curved arraysrequire a smaller number of optical elements and results in a largerfield of view.

Currently, studies that have investigated optical communication for UUVsare very limited and focus on planar arrays for Autonomous UnderwaterVehicles (AUVs). These studies include an estimation of AUV orientationto a beacon by using a photodiode array and distance measurement betweentwo UUVs. In addition to array designs for communication between UUVs,other studies have investigated optical communications for dockingoperations. For example, a single detector (quadrant photodiode) hasbeen used to operate as a 2×2 detector array. In addition, researchershave mounted an optical detector on an AUV to detect translationalmotion of the AUV with respect to a light source. Optical communicationfor distance sensing between a swarm of UUVs was conducted using a LEDtransceiver with an IrDA encoder/decoder chip. In addition to navigationpurposes, the use of optical communication has been investigated fortransmitting remote control commands and data transfer rates. Resultsbased on laboratory and field work showed that an optical modem systemconsisting of an omnidirectional light source and photomultiplier tubecan achieve a data streaming rate of up to 10 Mbit/s, with a reported1.2 Mbit/s data transfer rate up to 30 m underwater in clear waterconditions. Other studies utilized underwater sensor network consistingof static and mobile nodes for high-speed optical communication system,where a point-to-point node communication is proposed for data muling.

Previous studies using acoustic communication evaluated the controlperformance of the UUVs for docking applications, namely usingAutonomous Underwater Vehicles (AUVs) that include: Adaptive ControlStrategy Proportional-Integral-Derivative (PID);Multi-Input-Multi-Output controller; and Sliding Mode Controller (SMC)and its variants, namely High-Order SMC (HOSMC) and State DependentRiccati Equation-HOSMC (SDRE-HOSMC). Recent studies have demonstratedthe potential use of both acoustic and optical communication fordocking. In these systems, acoustic communication is used in relativelylonger ranges, 100 m, for navigating towards a docking station and videocameras are used in closer ranges, 8-10 m, to guide the vehicle into thedocking station. In this study, PID and SMC are investigated for bothstatic-dynamic and dynamic-dynamic systems.

The scope of some examples includes control between two UUVs and betweena UUV and docking stations) using primarily or only opticalcommunication. The work investigated control theory and ocean opticsconcepts that are used to develop models, algorithms and hardware. Threemain goals of this study are, for example:

1) Design of a cost-effective optical detector array interface. In orderto receive feedback to the controls, an optical detector array interfaceis vital. A guiding light beacon will be used as a transmitter. Thelight field intersecting with the detector module will be translatedinto an electronic signal for pose detection and control purposes.

2) Evaluation of control and image processing algorithms to be used inpose detection and UUV control. For timely and stable response of theUUV to the changes in the optical input coming from another UUV or adocking station, the performance of image processing and controlalgorithms need to be evaluated. The performance should take intoaccount the optical variability that exists in natural waters.

3) Development of optical detector hardware to obtain real-time posefeedback signal for the control of a UUV. A proof-of-concept hardwarewill demonstrate the performance of pose detection and control inlaboratory settings.

In addition to the goals of this study to develop an interface andcontrols between two UUVs and between a UUV and a docking station, thereare other applications that can benefit from this study, for example:

1) FSO communication—In this study a continuous-wave light source wasused as the transmitting signal. However, the bandwidth of thephotodetectors allows the transmission of pulsed signals which canprovide coded control signals and also data transfer.

2) Beam diagnostics—The two array designs, i.e. planar and curvedarrays, are compared based on their ability to generate a unique imagefootprint. This can also be used to evaluate scattering and absorptionof light through the water column in addition to the geometrical andenvironmental factors that affect the light travel in underwater.

UUV Modeling, Control and Stability Introduction

The control of a UUV to either navigate to a predefined point in spaceor to follow a path requires a fundamental understanding of the UUVmodel. In this chapter, UUV model is analyzed in two sections,kinematics, i.e. geometrical aspects of the motion without forceanalysis and UUV dynamics, i.e. analysis of the forces that contributeto the motion of the UUV. More detailed analysis of marine vehiclemodeling including UUVs is provided. This chapter summarizes the mainconcepts demonstrated in these sources.

UUV Kinematics

The UUV are capable of motion in 6 degrees-of-freedom (DOF) inunderwater. For analysis of the UUV motion, two coordinate frames areintroduced: 1) The moving coordinate frame, X_(o)Y_(o)Z_(o), which isfixed to the UUV body and thus also named body-fixed reference frame.X_(o) defines the longitudinal axis (aft to fore), Y_(o) defines thetransverse axis (port to starboard) and Z_(o) defines the normal axis(top to bottom). 2) Earth fixed reference frame. The motion of the UUVin body fixed frame is described in the earth fixed frame which is alsocalled inertial reference frame (FIG. 1A).

Because the rotation of the Earth does not affect the motion of the UUVssignificantly (as they are considered as low-speed vehicles), it isassumed that the accelerations of a point on the Earth fixed referenceframe can be neglected. Thus, position and orientation of the UUV can beexpressed in Earth-fixed frame while the linear and angular velocitiesare expressed in the body-fixed reference frame. The variables in thismanuscript are defined according to the SNAME (the Society of NavalArchitects and Marine Engineers) (1950) notation as demonstrated inTable 1.1.

TABLE 1.1 SNAME notation for marine vehicles Linear and Motion Forcesand angular Positions and DOF type Moments velocities Euler angles 1Surge X u x 2 Sway Y v y 3 Heave Z w z 4 Roll K p φ 5 Pitch M q θ 6 YawN r ψ

The motion of a UUV in 6-DOF can be represented in the followingvectorial forms:

η=[η₁ ^(T),η₂ ^(T)]^(T) η₁ =[x,y,z] ^(T) η₂=[φ,θ,ψ]^(T)  (1.1)

ν=[ν₁ ^(T),ν₂ ^(T)]^(T) ν₁ =[u,v,w] ^(T) ν₂ [p,q,r] ^(T)  (1.2)

τ=[τ₁ ^(T),ρ₂ ^(T)] τ₁ =[X,Y,Z] ^(T) τ₂ =[K,M,N] ^(T)  (1.3)

ηε

^(6×1) denotes the position and orientation in Earth-fixed coordinatesystem, σ×

^(6×1) denotes the linear and angular velocities acting on thebody-fixed frame and τε

^(6×1) represents the forces and the moments acting on the UUV on thebody-fixed reference frame. In this manuscript, the orientation isdescribed by Euler angles.

Euler Angles

The vehicle motion in body-fixed reference frame can be transformed intoEarth-fixed coordinate system through a velocity transformation as in

{dot over (η)}₁ =J ₁(η₂)ν₁  (1.4)

J₁(η₂) is the linear velocity transformation matrix from linearbody-fixed velocity vector to the velocities expressed in Earth-fixedreference frame. The transformation matrix is a function of roll (φ),pitch (θ) and yaw (ψ) angles. J₁(η₂) is described through a series ofrotation sequences (3-2-1) as follows:

J ₁(η₂)=C _(z,ψ) ^(T) C _(y,θ) ^(T) C _(x,φ) ^(T)  (1.5)

where the principle rotations C_(z,ψ) ^(T), C_(y,θ) ^(T), C_(x,φ) ^(T)are defined as

$\begin{matrix}{{C_{z,\psi}^{T} = {{\begin{bmatrix}{c\; \psi} & {s\; \psi} & 0 \\{{- s}\; \psi} & {c\; \psi} & 0 \\0 & 0 & 1\end{bmatrix}\mspace{14mu} C_{y,\theta}^{T}} = \begin{bmatrix}{c\; \theta} & 0 & {{- s}\; \theta} \\0 & 1 & 0 \\{s\; \theta} & 0 & {c\; \theta}\end{bmatrix}}}{C_{x,\varphi}^{T} = \begin{bmatrix}1 & 0 & 0 \\0 & {c\; \varphi} & {s\; \varphi} \\0 & {{- s}\; \varphi} & {c\; \varphi}\end{bmatrix}}} & (1.6)\end{matrix}$

Here c(•) and s(•) represent cosine and sine functions, respectively.Expanding (2.5) results in:

$\begin{matrix}{{J_{1}( \eta_{2} )} = \begin{bmatrix}{c\; \psi \; c\; \theta} & {{{- s}\; \psi \; c\; \varphi} + {c\; \psi \; s\; \theta \; s\; \varphi}} & {{s\; \psi \; s\; \varphi} + {c\; \psi \; c\; \varphi \; s\; \theta}} \\{s\; \psi \; c\; \theta} & {{c\; \psi \; c\; \varphi} + {s\; \varphi \; s\; \theta \; s\; \psi}} & {{{- c}\; \psi \; s\; \varphi} + {s\; \theta \; s\; \psi \; c\; \varphi}} \\{{- s}\; \theta} & {c\; \theta \; s\; \varphi} & {c\; \theta \; c\; \varphi}\end{bmatrix}} & (1.7)\end{matrix}$

Similarly, the angular velocities acting on the body-fixed frame can betransformed into Euler rate vector {dot over (η)}₂=[{dot over (φ)},{dotover (θ)},{dot over (ψ)}]^(T) as in

η₂ =J ₂(η₂)ν₂  (1.8)

J₂(η₂) is the angular velocity transformation matrix that transformsfrom angular body-fixed reference frame to Euler rate vector.Integration of Euler rate vector yields Euler angles. J₂(η₂) isexpressed as:

$\begin{matrix}{{J_{2}( \eta_{2} )} = \begin{bmatrix}1 & {s\; \varphi \; t\; \theta} & {c\; \varphi \; t\; \theta} \\0 & {c\; \varphi} & {{- s}\; \varphi} \\0 & {s\; {\varphi/c}\; \theta} & {c\; {\varphi/c}\; \theta}\end{bmatrix}} & (1.9)\end{matrix}$

where t(•) represents tangent function. It should be noticed that for apitch angle of θ=+90°, J₂(η₂) is undefined. Because UUVs can operateclose to this singularity point, this could present a problem. Thiscould be resolved by using quaternion representation [Fossen 1] ratherthan Euler angles. In this manuscript, the UUVs are assumed to bemechanically designed and built stable to be within θ=±10°. Therefore,it is mechanically prevented to be close to the singularity point.

UUV Dynamics

6-DOF nonlinear UUV dynamic equations are expressed as

M{dot over (ν)}+C(ν)ν+D(ν)ν+g(η)=τ  (1.10)

where Mε

^(6×6) is the inertial matrix including rigid body terms, M_(RB), andadded mass, M_(A). C(ν)ε

^(6×6) is the Coriolis and centripetal terms consisting of rigid bodyCoriolis and centripetal terms C_(RB) and hydrodynamic Coriolis andcentripetal terms, C_(A). D(ν)ε

^(6×6) is the damping force matrix, g(η)ε

^(6×1) is the gravitational forces and moments and τε

^(6×1) is the vector of control inputs. The UUV 6-DOF dynamic equationsare expressed using Newton's second law.

Newton-Euler Formulation

The foundations of Newton-Euler formulation are based on the Newton'ssecond law relating the mass, m, acceleration, {dot over (ν)}_(c), andforce, f_(c), as follows

m{dot over (ν)}=f _(c)  (1.11)

subscript denotes the center of mass of the body. Euler's first axiomstates that the linear momentum of a body, p_(c) is equal to the productof the mass and the velocity of the center of mass:

mν _(c) =p _(c)  (1.12)

Euler's second axiom states that the rate of change of angular momentum,h_(c), about a point fixed in Earth fixed reference frame or center ofmass of the body is equal to the sum of external torques:

I _(c) ω=h _(c)  (1.13)

where I_(c) is the inertia tensor about the center of gravity. Theseexpressions are used to derive the UUV rigid body equations of motion.

Rigid-Body Dynamics

Defining a body-fixed coordinate frame X_(o)Y_(o)Z_(o) rotating with anangular velocity vector ω=[ω₁,ω₂,ω₃]^(T), about an Earth-fixedcoordinate system XYZ, the inertia tensor of the body, I_(o), in thebody-fixed coordinate system X_(o)Y_(o)Z_(o) with an origin O is definedas

$\begin{matrix}{I_{o}\overset{\Delta}{=}\begin{bmatrix}I_{x} & {- I_{xy}} & {- I_{xz}} \\{- I_{yx}} & I_{y} & {- I_{yz}} \\{- I_{zx}} & {- I_{zy}} & I_{z}\end{bmatrix}} & (1.14)\end{matrix}$

I_(x), I_(y) and I_(z) are the moments of inertia about the X_(o), Y_(o)and Z_(o) axes while the products of inertia I_(xy)=I_(yx),I_(xz)=I_(zx) and I_(yz)=I_(zy). The elements of the inertia tensor aredefined as

I _(x)=∫(y ² +z ²)ρ_(A) dV; I _(xy) =∫xyρ _(A) dV=I _(yx)  (1.15)

I _(y)=∫(x ² +z ²)ρ_(A) dV; I _(xz) =∫xzρ _(A) dV=I _(zx)  (1.16)

I _(z)=∫(x ² +y ²)ρ_(A) dV; I _(yz) =∫yzρ _(A) dV=I _(yz)  (1.17)

ρ_(A) is the mass density of the bodyThe inertia tensor, I_(o), can be represented in the vectorial form as:

I _(o) ω=∫r×(ω×r)ρ_(A) dV  (1.18)

The mass of the body can be defined as

m=∫ρ _(A) dV  (1.19)

FIG. 1B shows the earth-fixed non-rotating reference frame XYZ and thebody-fixed rotating reference frame X_(o)Y_(o)Z_(o).

The underlying assumptions in the dynamics analysis of UUV are

-   -   1) The vehicle mass is constant in time, i.e. {dot over (m)}=0.    -   2) The vehicle is rigid: This assumption neglects the        interacting forces between the individual UUV parts.    -   3) The Earth-fixed reference frame is inertial: This assumption        eliminates the need to include the occurring forces due to        Earth's motion relative to a star-fixed reference system which        is used in space applications.

By utilizing the first assumption, the distance from the origin of thebody fixed reference frame, X_(o)Y_(o)Z_(o), to the vehicle's center ofgravity is

$\begin{matrix}{r_{G} = {\frac{1}{m}{\int{r\; \rho_{A}{V}}}}} & (1.20)\end{matrix}$

In order to obtain the equations of motion from a selected arbitraryorigin in the body-fixed coordinate system, the following formula isused

ċ=ċ _(B) +ω×c  (1.21)

This formula relates the time derivative of an arbitrary vector in theEarth-fixed frame, XYZ, i.e. ċ to the time derivative of an arbitraryvector in the body-fixed reference frame, X_(o)Y_(o)Z_(o), i.e. ċ_(B).This relation yields:

{dot over (ω)}={dot over (ω)}_(B)+ω×ω={dot over (ω)}_(B)  (1.22)

stating that the angular acceleration is equal in both reference frames.

Translational Motion

From FIG. 1B it is seen that the distance from the origin of theEarth-fixed reference frame to the center of gravity of the vehicle,i.e. r_(C) can be expressed as

r _(c) =r _(G) +r _(o)  (1.23)

Thus, the velocity of the center of gravity is

ν_(c) ={dot over (r)} _(c) ={dot over (r)} _(o) +{dot over (r)}_(G)  (1.24)

Utilizing the following relations ν_(o)={dot over (r)}_(o) and {dot over(r)}_(GB)=0 for rigid body,

{dot over (r)} _(G) ={dot over (r)} _(G) _(B) +ω×r _(G) =ω×r_(G)  (1.25)

{dot over (r)}_(G) _(B) stands for time-derivative with respect to thebody-fixed reference frame, X_(o)Y_(o)Z_(o). Therefore,

ν_(C)=ν_(o) +ω×r _(G)  (1.26)

The acceleration vector is:

{dot over (ν)}_(C)={dot over (ν)}_(o) +{dot over (ω)}×r _(G) +ω×{dotover (r)} _(G)  (1.27)

which in turn yields:

{dot over (ν)}_(C)={dot over (ν)}_(o) _(B) +{dot over (ω)}_(B) ×r_(F)+ω×(ω×r _(G))  (1.28)

Substituting (2.28) into (2.11) results in

$\begin{matrix}{{m( {{\overset{.}{v}}_{o_{B}} + {\omega \times v_{o}} + {{\overset{.}{\omega}}_{B} \times r_{G}} + {\omega \times ( {\omega \times r_{G}} )}} )} = f_{o}} & (1.29)\end{matrix}$

If the arbitrary origin of the body-fixed coordinate systemX_(o)Y_(o)Z_(o) is chosen to coincide with the center of gravity, thedistance from the center of gravity to the origin, r_(G)=[0,0,0]^(T) andwith f_(o)=f_(C) and ν_(o)=ν_(C), (2.29) reduces to

m({dot over (ν)}_(C) _(B) +ω×ν_(C))=f _(C)  (1.30)

Rotational Motion

The absolute momentum at the origin in FIG. 1B is defined as

h _(o)

∫r×νρ _(A) dV  (1.31)

Taking the time derivative of (2.31) yields:

{dot over (h)} _(o) =∫r×ρ _(A) dV+∫{dot over (r)}×νρ _(A) dV  (1.32)

Noticing that m_(o)=∫r×{dot over (ν)}ρ_(A)dV and ν={dot over(r)}_(o)+{dot over (r)} which implies {dot over (r)}=ν−ν_(o). Pluggingin these relations to (2.32) yields

{dot over (h)} _(o) m _(o)−ν_(o)×∫νρ_(A) dV  (1.33)

or

h _(o) =m _(o)−ν_(o)×∫(ν_(o) +{dot over (r)})ρ_(A) dV=m _(o)−ν_(o)×∫{dot over (r)}ρ _(A) dV  (1.34)

(2.34) can be rewritten by taking the time derivative of (2.20) as:

m{dot over (r)} _(G) =∫{dot over (r)}ρ _(A) dV  (1.35)

Using the fact that {dot over (r)}_(G)=ω×r_(G), (2.35) is rewritten as

∫{dot over (r)}ρ _(A) dv=m(ω×r _(G))  (1.36)

Substituting (2.36) in (2.34) results in

{dot over (h)} _(o) =m _(o) −mν _(o)×(ω×r _(G))  (1.37)

Writing (2.31) as

h _(o) =∫r×νρ _(A) dV=∫r×ν _(o)ρ_(A) dV+∫r×(ω×r)ρ_(A) dV  (1.38)

∫r×ν_(o)ρ_(A)dV term in (2.38) can be rewritten as

∫r×ν _(o)ρ_(A) dV=(∫rρ _(A) dV)×ν_(o) =mr _(G)×ν_(o)  (1.39)

(2.38) reduces to

h _(o) =I _(o) ω+mr _(G)×ν_(o)  (1.40)

Under the assumption that I_(o) is constant, we take the time-derivativeof (2.40):

{dot over (h)} _(o) =I _(o){dot over (ω)}_(B)+ω×(I _(o)ω)+m(ω×r_(G))×ν_(o) +mr _(G)×({dot over (ν)}_(o) _(B) +ω×ν_(o))  (1.41)

Using the relations (ω×r_(G))×ν_(o)=−ν_(o)×(ω×r_(G)) and eliminating{dot over (h)}_(o) term from (2.37) and (2.41) yields

I _(o){dot over (ω)}_(B)+ω×(I _(o)ω)+mr _(G)×({dot over (ν)}_(o) _(B)+ω×ν_(o))=m _(o)  (1.42)

If the origin of the body-fixed coordinate system X_(o)Y_(o)Z_(o) ischosen to coincide with the center of gravity of the UUV, then (2.42)simplifies to

I _(c)ω+ω×(I _(c)ω)=m _(C)  (1.43)

DOF Rigid-Body Equations of Motion

In this section, vectorial representation of the UUV dynamics will beshown. In addition, assumptions that simplify the equations of motionwill be introduced. Applying the following SNAME notation

f _(o)=τ₁ =[X,Y,Z] ^(T)=External Forces

m _(o)=τ₂ =[K,M,N] ^(T)=External Moments about origin O

v _(o)=ν₁ =[u,v,w] ^(T)=Linear velocities on body-fixed coordinate frameX _(o) Y _(o) Z _(o)

ω=ν₂ =[p,q,r] ^(T)=Angular velocities on body-fixed coordinate frame X_(o) Y _(o) Z _(o)

r _(G) =[x _(G) ,y _(G) ,z _(G)]^(T)=center of gravity

Applying this notation to the translational and rotational motionequations shown in previous sub-sections yields

$\begin{matrix}{\mspace{79mu} {{m\lbrack {\overset{.}{u} - {vr} + {wq} - {x_{G}( {q^{2} + r^{2}} )} + {y_{G}( {{pq} - \overset{.}{r}} )} + {z_{G}( {{pr} + \overset{.}{q}} )}} \rbrack} = X}} & (1.44) \\{\mspace{76mu} {{m\lbrack {\overset{.}{v} - {wp} + {ur} - {y_{G}( {r^{2} + p^{2}} )} + {z_{G}( {{qr} - \overset{.}{p}} )} + {x_{G}( {{pq} + \overset{.}{r}} )}} \rbrack} = Y}} & \; \\{\mspace{79mu} {{m\lbrack {\overset{.}{w} - {uq} + {vp} - {z_{G}( {p^{2} + q^{2}} )} + {x_{G}( {{pr} - \overset{.}{q}} )} + {y_{G}( {{qr} + \overset{.}{p}} )}} \rbrack} = Z}} & \; \\{{{I_{x}\overset{.}{p}} + {( {I_{z} - I_{y}} ){qr}} - {( {\overset{.}{r} + {pq}} )I_{xz}} + {( {r^{2} - q^{2}} )I_{yz}} + {( {{pr} - \overset{.}{q}} )I_{xy}} + {m\lbrack {{y_{G}( {\overset{.}{w} - {uq} + {vp}} )} - {z_{G}( {\overset{.}{v} - {wp} + {ur}} )}} \rbrack}} = K} & \; \\{{{{I_{y}\overset{.}{q}} + {( {I_{x} - I_{z}} ){pr}} - {( {\overset{.}{p} + {qr}} )I_{xy}} + {( {p^{2} - r^{2}} )I_{zx}} + {( {{pq} - \overset{.}{r}} )I_{yz}} + {m\lbrack {{z_{G}( {\overset{.}{u} - {vr} + {wq}} )} - {x_{G}( {\overset{.}{w} - {uq} + {vp}} )}} \rbrack}} = M}\mspace{551mu}} & \; \\{{{I_{z}\overset{.}{r}} + {( {I_{y} - I_{x}} ){pq}} - {( {\overset{.}{q} + {pr}} )I_{yz}} + {( {q^{2} - p^{2}} )I_{xy}} + {( {{rq} - \overset{.}{p}} )I_{zx}} + {m\lbrack {{x_{G}( {\overset{.}{v} - {wp} + {ur}} )} - {y_{G}( {\overset{.}{u} - {vr} + {wq}} )}} \rbrack}} = N} & \;\end{matrix}$

These equations can be represented in a more compact, vectorial form asfollows

M _(RB) ν+C _(RB)(ν)ν=τ_(RB)  (1.45)

The rigid-body equations can be simplified by choosing the origin of thebody fixed-coordinate frame coinciding with the center of gravity. Inthis case r_(G)=[0,0,0] and all the center of gravity related terms dropout of equation. This yields

m(u−vr+wq)=X I _(x) _(C) p+(I _(z) _(C) −_(y) _(C) )qr=K

m(v−wp+ur)=Y I _(y) _(C) q+(I _(x) _(C) −I _(z) _(C) )pr=M

m(w−uq+vp)=Z I _(z) _(C) r+(I _(y) _(C) −I _(x) _(C) )pq=N  (1.46)

Hydrodynamic Forces and Moments

Hydrodynamic forces acting on the rigid bod, are analyzed asradiation-induced forces, i.e. when the rigid body is forced tooscillate with the wave excitation frequency and there are no incidentwaves. In this case, the radiation induced forces and moments can beanalyzed in

-   -   1) Added mass due to inertia of the surrounding fluid    -   2) Damping effects due to potential damping, skin friction, wave        drift damping and vortex shedding    -   3) Restoring forces due to weight and buoyancy

The effect of hydrodynamic forces acting on the vehicle can be shown asfollows

τ_(H) =−M _(A) ν−C _(A)(ν)ν−D(ν)ν−g(η)  (1.47)

where M_(A) and C_(A)(ν) are added mass and hydrodynamic Coriolis andcentripetal term matrices, D(ν) is the hydrodynamic damping matrixincluding potential damping, skin friction, wave drift damping andvortex shedding, g(η) is the restoring forces. In addition to thehydrodynamic forces exerting on the UUV during the motion, environmentalforces also affect the UUV motion. The environmental forces are mainlydue to ocean currents, waves and winds. Combining all of these effects,the 6-DOF dynamic equations of motion of a UUV is

M{dot over (ν)}+C(ν)ν+D(ν)ν+g(η)=τ+τ_(H)+τ_(E)  (1.48)

where τ is the propulsion forces including the thruster/propellers andcontrol surfaces/rudder forces, τ_(E) denotes the environmental forces.

Added Mass

Added mass is the pressure-induced forces and moments due to a forcedharmonic motion of the body which are proportional to the accelerationof the body. As the UUV passes through the fluid, the fluid must moveaside and close behind the vehicle, i.e. open the passage for the UUV.The fluid passage possesses the kinetic energy which would be lacked ifthe UUV is stationary. Fluid kinetic energy can be written as

T _(A)=½ν^(T) M _(A)ν  (1.49)

M_(A) ε

^(6×6) is the added inertia matrix defined as

$\begin{matrix}{M_{A}\overset{\Delta \;}{=}{- \begin{bmatrix}X_{\overset{.}{u}} & X_{\overset{.}{v}} & X_{\overset{.}{w}} & X_{\overset{.}{p}} & X_{\overset{.}{q}} & X_{\overset{.}{r}} \\Y_{\overset{.}{u}} & Y_{\overset{.}{v}} & Y_{\overset{.}{w}} & Y_{\overset{.}{p}} & Y_{\overset{.}{q}} & Y_{\overset{.}{r}} \\Z_{\overset{.}{u}} & Z_{\overset{.}{v}} & Z_{\overset{.}{w}} & Z_{\overset{.}{p}} & Z_{\overset{.}{q}} & Z_{\overset{.}{r}} \\K_{\overset{.}{u}} & K_{\overset{.}{v}} & K_{\overset{.}{w}} & K_{\overset{.}{p}} & K_{\overset{.}{q}} & K_{\overset{.}{r}} \\M_{\overset{.}{u}} & M_{\overset{.}{v}} & M_{\overset{.}{w}} & M_{\overset{.}{p}} & M_{\overset{.}{q}} & M_{\overset{.}{r}} \\N_{\overset{.}{u}} & N_{\overset{.}{v}} & N_{\overset{.}{w}} & N_{\overset{.}{p}} & N_{\overset{.}{q}} & N_{\overset{.}{r}}\end{bmatrix}}} & (1.50)\end{matrix}$

For many UUV applications, the vehicle will be allowed to move at lowspeeds. If the vehicle is assumed to have three planes of symmetry, thenM_(A) and C_(A)(ν) simplifies to

$\begin{matrix}{M_{A} = {{- {diag}}\{ {X_{\overset{.}{u}},Y_{\overset{.}{v}},Z_{\overset{.}{w}},K_{\overset{.}{p}},M_{\overset{.}{q}},N_{\overset{.}{r}}} \}}} & (1.51) \\{{C_{A}(v)} = {- \begin{bmatrix}0 & 0 & 0 & 0 & {{- Z_{\overset{.}{w}}}w} & {Y_{\overset{.}{v}}v} \\0 & 0 & 0 & {Z_{\overset{.}{w}}w} & 0 & {{- X_{\overset{.}{u}}}u} \\0 & 0 & 0 & {{- Y_{\overset{.}{v}}}v} & {X_{\overset{.}{u}}u} & 0 \\0 & {{- Z_{\overset{.}{w}}}w} & {Y_{\overset{.}{v}}v} & 0 & {{- N_{\overset{.}{r}}}r} & {M_{\overset{.}{q}}q} \\{Z_{\overset{.}{w}}w} & 0 & {{- X_{\overset{.}{u}}}u} & {N_{\overset{.}{r}}r} & 0 & {{- K_{\overset{.}{p}}}p} \\{Y_{\overset{.}{v}}v} & {X_{\overset{.}{u}}u} & 0 & {{- M_{\overset{.}{q}}}q} & {K_{\overset{.}{p}}p} & 0\end{bmatrix}}} & (1.52)\end{matrix}$

Strip Theory

In order to estimate the hydrodynamic derivatives, i.e. added inertiamatrix terms, strip theory is used. By dividing the submerged part ofthe vehicle into strips, the hydrodynamic coefficients can be computedfor each strip and estimated over the length of the UUV to obtainthree-dimensional results. For submerged slender vehicles the followingformulas can be used to obtain the hydrodynamic coefficients

−X _(u)=∫_(−L/2) ^(L/2) A ₁₁ ^((2D))(y,z)dx≈0.10 m  (1.53)

−Y _(v)=∫_(−L/2) ^(L/2) A ₂₂ ^((2D))(y,z)dx  (1.54)

−Z _(w)=∫_(−L/2) ^(L/2) A ₃₃ ^((2D))(y,z)dx  (1.55)

−K _(p)=∫_(−L/2) ^(L/2) A ₄₄ ^((2D))(y,z)dx  (1.56)

−M _(q)=∫_(−L/2) ^(L/2) A ₅₅ ^((2D))(y,z)dx  (1.57)

−N _(r)=∫_(−L/2) ^(L/2) A ₆₆ ^((2D))(y,z)dx  (1.58)

where L is the length of the vehicle. A₂₂ ^((2D)), A₃₃ ^((2D)) and A₄₄^((2D)) values are approximated using the values in FIG. 1C depending onthe UUV body type.

FIG. 1C shows two dimensional added mass coefficients used in striptheory. Two-dimensional hydrodynamic coefficients for roll, pitch andyaw angles can be found by

∫_(−L/2) ^(L/2) A ₄₄ ^((2D))(y,z)dx

∫ _(−B/2) ^(B/2) y ² A ₃₃ ^((2D))(x,z)dy+∫ _(−H/2) ^(H/2) z ² A ₂₂^((2D))(x,y)dz  (1.59)

∫_(−L/2) ^(L/2) A ₅₅ ^((2D))(y,z)dx

∫ _(−L/2) ^(L/2) x ² A ₃₃ ^((2D))(x,z)dx+∫ _(−H/2) ^(H/2) z ² A ₁₁^((2D))(x,y)dz  (1.60)

∫_(−L/2) ^(L/2) A ₆₆ ^((2D))(y,z)dx

∫ _(−B/2) ^(B/2) y ² A ₁₁ ^((2D))(x,z)dy+∫ _(−H/2) ^(H/2) z ² A ₁₁^((2D))(x,y)dx  (1.61)

B and H are the width and height of the vehicle. For other geometricaltypes of vehicles, more detailed analysis can be found. Another approachto estimate the hydrodynamic coefficients is to use hydrodynamiccomputation software such as, WAMIT, RESPONSE, SIMAN, MIMOSA, SIMO andWAVERES, etc.

Hydrodynamic Damping

Hydrodynamic damping for marine vehicles is sometimes mainly caused bypotential damping, skin friction, wave drift damping and vortexshedding.

Potential damping is the radiation induced damping term encountered whenthe UUV body is forced to oscillate with the wave excitation frequencyin the absence of incident waves. The contribution from potentialdamping is negligible in comparison to the dissipative terms likeviscous damping.

Skin friction is due to laminar boundary layer when the vehicleundergoes low-frequency motion. In addition to the linear skin friction,there is also quadratic skin friction effects that should be taken intoaccount during the design of the control system.

Wave-drift damping is the added resistance for surface vessels. Thus, itis not a dominant affect for UUVs.

Vortex Shedding is caused by frictional forces in a viscous fluid. Theviscous damping force due to vortex shedding can be formulated as:

f(U)=−½ρC _(D)(R _(n))A|U|U  (1.62)

where U is the vehicle speed, ρ is the surrounding water density, A isthe projected cross-sectional area in water, C_(D) (R_(n)) is the dragcoefficient as a function of Reynolds number as

$\begin{matrix}{R_{n} = \frac{UD}{v}} & (1.63)\end{matrix}$

where D is the characteristic length of the vehicle and ν is thekinematic viscosity coefficient (for salt water at 5° C. with salinity3.5%, ν=1.56·10⁻⁶). Quadratic drag in 6-DOF is expressed as:

$\begin{matrix}{{D_{M}{\text{(}\text{v}\text{)}}v} = \begin{bmatrix}{{v}^{T}D_{1}\nu} \\{{v}^{T}D_{2}\nu} \\{{v}^{T}D_{3}\nu} \\{{v}^{T}D_{4}\nu} \\{{v}^{T}D_{5}\nu} \\{{v}^{T}D_{6}\nu}\end{bmatrix}} & (1.64)\end{matrix}$

D_(i) (i=1 . . . 6)ε

^(6×6) depend on ρ, C_(D) and A. Thus each term in (2.64) is different.Subscript M in D_(M) stands for Morison's equation.

Restoring Forces and Moments

The gravitational and buoyancy forces actin on the marine vehicle arenamed restoring forces in the hydrodynamic terminology. f_(G), thegravitational force, acts on the center of gravity r_(G)=[x_(G), y_(G),z_(G) ^(T)] while f_(B), the buoyancy force, acts on the center ofbuoyancy r_(B)=[x_(B), y_(B), z_(B)]^(T). For underwater vehicles,defining m as the mass of the vehicle and ∇ as the volume of fluiddisplaced by the vehicle, g the gravitational acceleration and ρ as thefluid density, the submerged weight of the body is W=mg, and thebuoyancy force is B=ρg∇. The weight and the buoyancy force can betransformed in the body-fixed coordinate frame as

$\begin{matrix}{{{{f_{G}( \eta_{2} )} = {{J_{1}^{- 1}( \eta_{2} )}\begin{bmatrix}0 \\0 \\W\end{bmatrix}}}\mspace{20mu} {{f_{B}( \eta_{2} )} = {- {{J_{1}^{- 1}( \eta_{2} )}\begin{bmatrix}0 \\0 \\B\end{bmatrix}}}}}} & (1.65)\end{matrix}$

The restoring force and moment vector can be expressed as:

$\begin{matrix}{{g(\eta)} = {- \begin{bmatrix}{{f_{G}(\eta)} + {f_{B}(\eta)}} \\{{r_{G} \times {f_{G}(\eta)}} + {r_{B} \times {f_{B}(\eta)}}}\end{bmatrix}}} & (1.66)\end{matrix}$

Expanding this expression results in

$\begin{matrix}{{g(\eta)} = \begin{bmatrix}{( {W - B} )s\; \theta} \\{{- ( {W - B} )}c\; \theta \; s\; \varphi} \\{{- ( {W - B} )}c\; {\theta c\varphi}} \\{{{- ( {{y_{G}W} - {y_{B}B}} )}c\; {\theta c\varphi}} + {( {{z_{G}W} - {z_{B}B}} )c\; {\theta s\varphi}}} \\{{( {{z_{G}W} - {z_{B}B}} )s\; \theta} + {( {{x_{G}W} - {x_{B}B}} )c\; {\theta c\varphi}}} \\{{{- ( {{x_{G}W} - {x_{B}B}} )}c\; {\theta s\varphi}} + {( {{y_{G}W} - {y_{B}B}} )\; s\; \theta}}\end{bmatrix}} & (1.67)\end{matrix}$

which is the Euler angle representation of the hydrostatic forces andmoments. If the UUV is neutrally buoyant, then W=B. Defining thedistance between the center of gravity r_(G) and the center of buoyancyr_(B) as:

BG=[BG _(x) ,BG _(y) ,BG _(z)]^(T) =[x _(G) −x _(B) ,y _(G) −y _(B) ,z_(G) −z _(B)]^(T)  (1.68)

Therefore, (2.67) simplifies to

$\begin{matrix}{{g(\eta)} = \begin{bmatrix}0 \\0 \\0 \\{{{- {\overset{\_}{BG}}_{y}}{Wc}\; {\theta c}\; \varphi} + {{\overset{\_}{BG}}_{z}{Wc}\; {\theta s}\; \varphi}} \\{{{\overset{\_}{BG}}_{z}{Ws}\; \theta} + {{\overset{\_}{BG}}_{x}{Wc}\; {\theta c}\; \varphi}} \\{{{{- {\overset{\_}{BG}}_{x}}{Wc}\; {\theta s}\; \varphi} - {{\overset{\_}{BG}}_{y}{Ws}\; \theta}}\;}\end{bmatrix}} & (1.69)\end{matrix}$

UUV Controllers and Stability

Proportional-Integral-Derivative (PID) Control

Most UUV systems, specifically Remotely Operated Vehicles (ROVs) utilizea series of single-input-single-output (SISO) PID controllers to controleach DOF. This suggests the use of the control gain matrices K_(p),K_(i) and K_(d) in the PID control law as follows:

τ_(PID) =K _(p) e(t)+K _(d) e(t)+K _(i) ∫e(τ)dτ  (1.70)

e=η_(d)−η is the tracking error, η_(d) denotes the vector of desiredstates and η denote the vector of measured states from the sensors.Throughout this manuscript, η is the pose output obtained from theoptical detector array (FIG. 1D). FIG. 1D shows a UUV Control Blockdiagram with the output obtained from optical feedback array. Controllerregulates the UUV motion based on the feedback obtained from the opticaldetector array and changes the course of the UUV by sending commands tothe thrusters.

The control problems for the static-dynamic and dynamic-dynamic casesdemonstrated in this manuscript can be evaluated as a set-pointregulation problem in which the desired state vector η_(d) is constant.In the static-dynamic case, the UUV navigates to a position based on theguidance obtained from the static light source. In the dynamic-dynamiccase the follower UUV follows the changing path of the leader UUV withthe desired state vector staying constant.

PID Stability for UUVs

In the set point regulation problem, PID controller of a nonlinearsquare system is shown to guarantee local stability as follows: Thegeneralized momentum, p, of the UUV is

p=M _(η){dot over (η)}  (1.71)

where M_(η) is the mass represented in the Earth-fixed coordinatesystem. The inertia matrix M represents the mass with respect to thebody-fixed coordinate system such that

M _(η) =J ^(−T)(η)MJ ⁻¹(η)  (1.72)

where J is the transformation matrix relating the body and Earth-fixedcoordinate systems (as discussed previously). A PID control law is takento be of the following form:

u=B ⁻¹ [J ^(T)(η)(K _(p) e+K _(i)∫₀ ^(t) e(τ)dτ−K _(d){dot over(η)})+g(η)]  (1.73)

In addition, a Lyapunov function candidate is given as

$\begin{matrix}{{V(x)} = {\frac{1}{2}{x^{T}\begin{bmatrix}M_{\eta}^{- 1} & {\alpha I} & 0 \\{\alpha I} & K_{p} & K_{i} \\0 & K_{i} & {\alpha K}_{i}\end{bmatrix}}}} & (1.74)\end{matrix}$

α is a small positive constant and x is given as:

x=[pη∫ ₀ ^(t) e(τ)dτ] ^(T)  (1.75)

Then, it has been shown that {dot over (V)}≦0 and η converges to aconstant η_(d). The PID controller parameters K_(p), K_(i) and K_(d) arematrices that satisfy:

$\begin{matrix}{K_{d} > M_{\eta}} & (1.76) \\{K_{i} > 0} & (1.77) \\{K_{p} > {Κ_{d} + {\frac{2}{\alpha}K_{i}}}} & (1.78)\end{matrix}$

Positive constant α is chosen such that it satisfies the followingcondition:

$\begin{matrix}{{{\frac{1}{2}( {1 - \alpha} )K_{d}} - {\alpha M}_{\eta} + {\frac{\alpha}{2}{\sum\limits_{i = 1}^{6}{( {\eta_{i} - \eta_{id}} )\; \frac{\partial m_{\eta}}{\partial\eta_{i}}}}}} > 0} & (1.79)\end{matrix}$

More details of the proof were given.

Sliding Mode Controller (SMC)

The dynamics of a UUV system can easily change when, for example, newsensor packages and tools are mounted on a UUV. The Sliding Mode Control(SMC) is a robust nonlinear control technique that is designed toaddress modeling uncertainties and has been employed in dynamicpositioning of remotely operated vehicles. The SMC, however, requires apriori knowledge of uncertainty bounds and assumes full-state feedback.In this study, SMC uses pose detection via image moment invariantsalgorithm for full state sensor feedback described in Chapter 4.

The SMC needs both position and velocity signals as inputs. In thedeveloped system, the detector array can provide position/orientationinputs directly to the controllers. However, for the velocity signals,the first derivatives of the pose signals are taken.

Because UUV motion in this study is restricted to be decoupled, SISOsystem approach is taken for UUV SMC system design. Therefore, fivesecond-order controllers are designed rather than a single fifth-ordercontroller:

x ^(n) =b(X;t)[ƒ(X;t)+U+d(t)]  (1.80)

where, x^(n) is the n^(th) derivative of state x, U is the control inputgenerated by the UUV propellers, d(t) is the potential disturbances suchas wave and currents, X=[x, {dot over (x)}, . . . , x^(n-1)]^(T) is thestate vector of the system (i.e., position, velocity and acceleration ofthe vehicle in a specific axis). ƒ(X; t) represents all lumped nonlinearfunctions in the system dynamics. For the follower UUV model used inthis research, ƒ(X; t) includes the velocity-dependent effects includingdrag forces and inertia. For a second order system, b(X; t) is theinverse of the inertia.The following simplified UUV model is used for pose detection for each5-DOF of under interest for this study:

m{umlaut over (x)}+c{dot over (x)}|{dot over (x)}|=u  (1.81)

where x is the state variable, m is the mass/inertia term (which alsoincludes added mass/inertia), c is the drag coefficient and u is thecontrol input.

A time-varying surface S(t) in the state space R^(n) is defined by thescalar equation s(X; t)=0 as in

$\begin{matrix}{{s( {X;t} )} = {( {\frac{\;}{t} + \lambda} )^{n - 1}\overset{\sim}{x}}} & (1.82)\end{matrix}$

λ is a positive constant and tracking error {tilde over (x)} is definedsuch that {tilde over (x)}=x−x_(d), where x_(d) denotes the desiredstate value. For a second order system (i.e., n=2), the sliding surfacebecomes

s(X;t)={tilde over ({dot over (x)})}+λ{tilde over (x)}  (1.83)

where s is a weighted sum of position and velocity errors. s(X; t)corresponds to a line that moves with the point (x_(d), {dot over(x)}_(d)) having a slope A.

The sliding condition is achieved when {dot over (s)}=0, where the errortrajectory {tilde over (x)} converges to the origin. For this, thederivative of the sliding surface is analyzed:

{dot over (s)}={umlaut over (x)}−{umlaut over (x)} _(d)+λ{tilde over({dot over (x)})}  (1.84)

The follower UUV model is represented in (28)

$\begin{matrix}{{\overset{.}{s} = {{{- \frac{c}{m}}\overset{.}{x}{\overset{.}{x}}} + {\frac{1}{m}u} - {\overset{¨}{x}}_{d} + {\lambda \overset{.}{\overset{\sim}{x}}}}}\;} & (1.85)\end{matrix}$

Setting {dot over (s)}=0 and combining (28) and (24), an equivalentcontrol law û may be obtained to help achieve the sliding condition {dotover (s)}=0 such that

û={circumflex over (m)}({umlaut over (x)} _(d)−λ{tilde over ({dot over(x)})})+c{dot over (x)}|{dot over (x)}|  (1.86)

In order to satisfy the sliding condition, a discontinuous term acrossthe surface s=0 is added to the û term such that

u=û−k1(s)  (1.87)

where 1(s) is a switching function and can be any odd function.Typically the signum function is used, but for this research, 1(s) ischosen to be the saturation function, sat(s/Φ), to eliminate the highfrequency chattering that is inherent in the signum function andundesirable for UUV thruster actuation. (Here, Φ represents the boundarylayer thickness of the switching function within which the switchingfunction is smooth and linear.) The discontinuous switching gain, k, ischosen to be larger than the maximum bounded uncertainty such that

k(x)=(F+βη)+{circumflex over (m)}(β−1)|{umlaut over (x)} _(d)−λ{tildeover ({dot over (x)})}|  (1.88)

where F is the estimation error bound on the nonlinear dynamics f, i.e.|{circumflex over (f)}−f|≦F. β is the gain margin of the system, definedas

$\begin{matrix}{\beta = \sqrt{\frac{b_{\max}}{b_{\min}}}} & (1.89)\end{matrix}$

where b_(min) and b_(max) are the minimum and maximum bounds on thecontrol gain b in the system, i.e. {umlaut over (x)}=f+bu. η is astrictly positive constant. In order to fully utilize the availablecontrol bandwidth, the control law is smoothed out in a time-varyingthin boundary layer

k (x)=k(x)−{dot over (Φ)}  (1.90)

where Φ is the boundary layer thickness. Tuning Φ to represent afirst-order filter of bandwidth, λ

k(x _(d))={dot over (Φ)}+λΦ  (1.91)

Setting the gain margin, β_(d)=β, the switching term with time-varyingthin boundary layer, k(x) is expressed as

$\begin{matrix}{{\overset{\_}{k}(x)} = {{k(x)} - {k( x_{d} )} + \frac{\lambda\Phi}{\beta_{d}}}} & (1.92)\end{matrix}$

Finally, the resulting control input u is

u=û−k (x)sat(s/Φ)  (1.93) SMC Stability for UUVs

A SMC for a Multiple Input Multiple Output (MIMO) UUV 6-DOF system isshown to be stable in the sense of Lyapunov as follows: Defining aLyapunov-like function candidate

V(S,t)=½s ^(T) M*s  (1.93)

where M*=J^(−T)MJ⁻¹. The time derivative of the Lyapunov function is

{dot over (V)}=s ^(T) M*{dot over (s)}+½s ^(T) {dot over (M)}*s−s ^(T)C*s+s ^(T) C*s  (1.94)

Incorporating the 6-DOF nonlinear UUV equations of motion as in (1.81)and assuming that the number of control inputs is equal to or more thannumber of controllable DOF:

{dot over (V)}=−s ^(T)(D*+K _(D))s+(J ⁻¹ s)^(T)({tilde over (M)}{umlautover (q)} _(r) +{tilde over (C)}{dot over (q)} _(r) +{tilde over(D)}{dot over (q)} _(r) +{tilde over (g)})−k ^(T) |J ⁻¹ s|  (1.95)

where {tilde over (M)}={circumflex over (M)}−M, {tilde over (C)}=Ĉ−C,{tilde over (D)}={circumflex over (D)}−D, {tilde over (g)}=ĝ−g and q_(r)denotes a virtual reference vector. Defining the switching term k_(i) as

k _(i) ≧|{tilde over (M)}{umlaut over (q)} _(r) +{tilde over (C)}({dotover (q)}){dot over (q)} _(r) +{tilde over (D)}({dot over (q)}){dot over(q)} _(r) +{tilde over (g)}(x)|_(i)+η_(i),  (1.96)

where η_(i)>0 as defined previously. This yields:

{dot over (V)}≦−s ^(T)(D*+K _(D))s−η ^(T)(J ⁻¹ s)≦0  (1.97)

The dissipative matrix D>0 and the gain matrix K_(D)≧0, resulting in(J^(−T)DJ⁻¹+K_(D))>0.

Characterization of Optical Communication in a Leader-Follower UUVFormation

As part of the research to development an optical communication designof a leader-follower formation between unmanned underwater vehicles(UUVs), this chapter presents light field characterization and designconfiguration of the hardware required to allow the use of distancedetection between UUVs. The study specifically is targetingcommunication between remotely operated vehicles (ROVs). As an initialstep in this study, the light field produced from a light source mountedon the leader UUV was empirically characterized and modeled. Based onthe light field measurements, a photo-detector array for the followerUUV was designed. Evaluation of the communication algorithms to monitorthe UUV's motion was conducted through underwater experiments in theJere E. Chase Ocean Engineering Laboratory at the University of NewHampshire. The optimal spectral range was determined based on thecalculation of the diffuse attenuation coefficients by using twodifferent light sources and a spectrometer. The range between the leaderand the follower vehicles for a specific water type was determined. Inaddition, the array design and the communication algorithms weremodified according to the results from the light field.

Preliminary work for this study included the development of a controldesign for distance detection of UUV using optical sensor feedback in aLeader-Follower formation. The distance detection algorithms weredesigned to detect translational motion above water utilizing a beam oflight for guidance. The light field of the beam was modeled using aGaussian function as a first-order approximation. This light field modelwas integrated into non-linear UUV equations of motion for simulation toregulate the distance between the leader and the follower vehicles to aspecified reference value. A prototype design of a photo-detector arrayconsisting of photodiodes was constructed and tested above water.However, before an array can be mounted on the bow of the follower UUV,a better understanding of the underwater light is needed. The proposedsystem is based on detecting the relative light intensity changes on thephotodiodes in the array. The size of the array strictly depends on thesize of the ROV. This chapter provides an overview on the experimentsand simulations conducted to adjust the algorithms based on underwaterconditions. Underwater light is attenuated due to the opticalcharacteristics of the water, which are constantly changing and are notuniformly distributed. As a result, applying distance detectionalgorithms underwater adds complexity and reduces operational ranges.Accordingly, the operation distance between the UUVs was limited to arange between 4.5 to 8.5 m for best performance. Experimental work inthis study was performed in the wave and tow tank at the Jere E. ChaseOcean Engineering facilities.

Theoretical Background

The basic concept for optical communication in this chapter is based onthe relative intensity measured between the detectors within thephoto-detector array mounted on the follower ROV. In addition to thebeam pattern produced by the light source, the intensity of lightunderwater follows two basic concepts in ocean optics: The inversesquare law and Beer-Lambert law.

Beam Pattern

The light field emitted from a light source can be modeled withdifferent mathematical functions. In addition, there are a variety oflight sources that can be used underwater that differ in their spectralirradiance (e.g., halogen, tungsten, and metal-halide). The spectralcharacteristics of the light source are an important issue that affectsthe illumination range, detector type and the detection algorithms. Asin the case of the light sources, the photo-detectors also have aspectral width in which their sensitivity is at a maximum value. Bydetermining the spectral characteristics of the light source, it ispossible to select the detector and filters for the photodetector array.We assume that the beam pattern can be modeled using a Gaussianfunction. This representation is valid for a single point light source.The Gaussian model used in this study can be represented as follows:

I(θ)=A*exp(−B*θ _(i) ²)  (2.1)

In (2.1), I is the intensity at a polar angle, θ_(i), where the originof the coordinate system is centered around the beam direction of thelight source. A and B are constants that describe the Gaussian amplitudeand width respectively.

Inverse Square Law

According to the inverse square law, the intensity of the light isinversely proportional to the inverse square of the distance:

$\begin{matrix}{I = \frac{S}{4\; {\pi r}^{2}}} & (2.2)\end{matrix}$

where I is the intensity at r distance away from the source and S is thelight field intensity at the surface of the sphere. Thus, the ratio ofthe light intensities at two different locations at the same axis can beexpressed as:

$\begin{matrix}{\frac{I_{1}}{I_{2}} = {\frac{\frac{S}{4\; {\pi r}_{1}^{2}}}{\frac{S}{4\; {\pi r}_{2}^{2}}} = ( \frac{r_{2}}{r_{1}} )^{2}}} & (2.3)\end{matrix}$

The light field S generated by a light source is assumed to show uniformillumination characteristics in all directions. In addition, the lightintensity is such that the light source is assumed to be a point sourceand that its intensity is not absorbed by the medium. It should also benoted that although the inverse square law is the dominant concept inthe development of control algorithms, for this research this is not theonly dominant optical mechanism that affects the light passing in water.As the light travels through water, its rays also get absorbed by themedium according to Beer-Lambert law.

Beer-Lambert Law

Beer-Lambert law states that radiance at an optical path length l in amedium decreases exponentially depending on the optical length, l, theangle of incidence, θ_(i), and the attenuation coefficient, K:Beer-Lambert law describes the light absorption in a medium under theassumption that an absorbing, source free medium is homogeneous andscattering is not significant. When the light travels through a medium,its energy is absorbed exponentially as in

$\begin{matrix}{{L( {\zeta,\hat{\xi}} )} = {{L( {0,\hat{\xi}} )}{\exp ( {- \frac{\zeta}{\mu}} )}}} & (2.4)\end{matrix}$

where L denotes the radiance, ζ the optical depth, {circumflex over (ξ)}the direction vector, and μ denotes the light distribution as a functionof angle θ such that:

μ=cos θ_(i)  (2.5)

Defining a quantity l, (i.e. the optical path length in direction μ),

$\begin{matrix}{{{dl} \equiv \frac{d\; \zeta}{\mu}} = \frac{{K(z)}{dz}}{\mu}} & (2.6)\end{matrix}$

where K(z) is the total beam attenuation coefficient and dz is thegeometric depth.

In this chapter, the experimental setup is built such that the incidenceangle θ_(i) is zero. As a result, combination of (2.4) and (2.6) resultsin:

L(ζ,{circumflex over (ξ)})=L(0,{circumflex over (ξ)})exp(−K(z)dz)  (2.7)

Experimental Setup

Experimental work in this study was performed in order to evaluate aproposed hardware design which was based on ocean optics and thehardware restrictions for a given ROV system. The experiments includedbeam diagnostics, spectral analysis and intensity measurements fromseveral light sources. These experiments were conducted in the Tow andWave Tank at the Ocean Engineering facilities. The wave and tow tank hasa tow carriage that moves on rails. A light source was mounted on arigid frame to the wall in the tow tank and a light detector was placedunderwater connected to a tow carriage (FIG. 2A). This experimentalsetup is based on the design. To characterize the interaction betweenthe light source and the light array a 50 Watt halogen lamp powered by12 V power source is used. For the detector unit, a spectrometer (OceanOptics Jaz) was used to characterize the underwater light field. Theseempirical measurements were used to adjust the detection algorithms andwill be also used in the design of the photo-detector array. The lightsource in the tank simulates a light source that is mounted on the crestof the leader ROV. The design of the photo-detector array simulates thearray that will be mounted on the bow of the follower ROV. Thephoto-detector array design depends on the size of the ROV and the lightfield produced by the light source mounted on the leader ROV. In thiscase, the size for an optical detector module was kept at 0.4 m, whichis the width dimension of the UNH ROV as a test platform, a smallobservation class ROV.

FIG. 2A shows an experimental schematic of UNH tow tank [73].Translational experiments in 1-D and 3-D (i.e., motion along andperpendicular to the center beam of the light source) were conducted inair and in water. The goals for the 1-D experiments were to characterizethe spectral properties of the water and to determine the best spectralranges for optical communication between the ROVs. In the underwaterexperiment, a submerged fiber optic cable with a collimator wasconnected to the spectrometer and was vertically aligned based on thepeak value of radiance emitted from the light source. This alignment isconsidered the illumination axis (x-axis). The radiance emitted from thelight source through the water column was empirically measured by thespectrometer at distances ranging from 4.5 to 8.5 m at 1 m increments.The spectrometer was configured to average 40 samples with anintegration time of 15 milliseconds. A 2° collimator was used torestrict the field of view collected by the spectrometer and to avoidthe collection of stray light rays reflecting off the tank walls or fromthe water surface.

The experimental setup in air was very similar, where the spectrometerwas mounted on a tripod and aligned to the peak value of radiance, theillumination axis (z-axis). Because such light sources produce heat athigh temperatures (up to 700° C.) that can damage the waterproof fixing,the experimental setup in air required that the light source besubmerged in an aquarium during operation. Similar to the underwaterexperiments, the same distances between the light source and thespectrometer, including the offsets, were maintained.

The 3-D translational underwater experiments utilized the same setup asthat of the underwater 1-D experiments where additional radiancemeasurements were conducted along a normal axis (z-axis) that is locatedon a plane normal to the illumination axis (x-axis). The 3-Dtranslational experiment maintained the same distances along theillumination axis between the light source and the spectrometer (i.e.,4.5 to 8.5 m), where additional measurements were conducted along thenormal axis at 0.1 m increments ranging from 0 m to 1 m. As mentionedpreviously, it is assumed that the light source is producing a beampattern that can be modeled using a Gaussian function. Accordingly, weassume that the radiance measurements along the normal axis aresymmetric in all directions. The experimental setups for 3-D underwaterexperiments can be seen in FIG. 2B. FIG. 2B shows an experimental setupfor translational 3-D underwater experiments. In particular, the leftimage shows a detector unit that includes a submerged fiber optic cablewith a collimator that was connected to the spectrometer and the rightimage shows a transmitting unit mounted to the wall of the tank.

Results

Light attenuation underwater causes a significant loss of radiant energyover increasing distances. The diffuse attenuation coefficient, K, isused as a parameter to calculate the decreased amount of energy from thelight source to the target. In this study, the diffuse attenuationcoefficient is used to determine the spectral range of the light sourceand determine the photo-detector types that can be utilized in thearray. For successful optical communication up to ranges of 9 m, theideal spectral ranges should be maintained such that the diffuseattenuation coefficient values are smaller than 0.1 m⁻¹. At thisdistance, the signal loses about half its energy. As a first-orderapproximation, the diffuse attenuation coefficient values are assumedconstant throughout the water column. This assumption reduces the numberof parameters used in the distance detection algorithms and theprocessing time used in future controls applications. In the study, thediffuse attenuation coefficient values are calculated for a 50 W lightsource. In FIG. 2C, the percentage loss curve for various distances isshown.

Diffuse attenuation was calculated based on (2.7). Measurements taken ata specific distance in water and in air are compared in order to accountfor the inverse square law. The light that travels in air also undergoesdiffuse attenuation but it is ignored in this case. These values suggestthat the UNH wave tank, where the experiments were conducted, containsalgae and dissolved matter. The study results suggest that 500-550 nmband-pass filters in the range should be used in the detector unit toprovide better performance of the distance detection algorithms.

FIG. 2C shows a percent attenuation graph. This graph shows the lightpercent attenuation per meter. It is seen that the spectral rangebetween 500-550 nm undergoes the least attenuation at any givendistance.

Based on the light attenuation results, the distance between the leaderand the follower vehicles was calculated. The experimental results (FIG.2.4) show that the performance of the algorithms in the UNH water tankis expected to decrease after 8.5 m. Beyond this range, the lightintensity falls into the background noise level (i.e., <20%).

FIG. 2D is an intensity vs. distance plot. The intensity readings arecollected between 500-550 nm and averaged. In this plot, theexperimental values are compared with the theoretical. Blue diamondsrepresent the experimental data and the green triangles represent thetheoretical calculations from taking the inverse square and Beer-Lambertlaws. The readings are normalized. The measurement at 4.5 m was used asthe reference measurement to normalize the intensity.

The light profile calculated from the 3-D experiments agrees with theassumption that the pattern of the light beam can be described using a2-D Gaussian fit (FIG. 2E). Using a 50% intensity decrease as athreshold, the effective beam radius from the center (i.e., theillumination axis) is 0.3 m. Another key finding obtained from the 3-Dexperiments, is the dimensions of the light detector array. It can beseen that if the length of the array is kept at 0.6 m, then differentlight detector elements can detect the light intensity change, which isuseful information for control algorithms. It should be stated that thephysical characteristics of the photo-detector array such as dimensionsand the spacing between the array elements strictly depend on beamdivergence.

FIG. 2E is a plot of the cross-sectional beam pattern. The measurementswere collected from 0 to 1.0 m at x-axis and at 4.5 m at theillumination axis for 50 W light source. The measurements between500-550 nm are averaged.

Discussion

The study results provide valuable environmental information formodifying a photo-detector array design according to light field.According to the diffuse attenuation, a 500-550 nm band-pass filter willallow the observation at the light field from a single source as a 2-DGaussian beam pattern. At this spectral range is around 0.1 m⁻¹ the peakpower of the beam (along the z-axis) will change from 100% to 23% as thearray moves away from light from 4.5 m to a distance of 8.5 m. The sizeof the beam pattern is a function of the divergence angle of the beam.In the current configuration, the Full width at half maximum (FWHM)radius expands from 0.3 m to 0.4 m as the array moves away from lightfrom 4.5 m to a distance of 8.5 m. The beam divergence can be modifiedusing reflectors and optic elements in case more acute changes in thelight field are needed over a shorter distance of 0.4 m, the maximumlength of the array.

During the empirical measurements in the UNH Tow Tank depth, severalerror sources were identified that limited an accurate correlationbetween the models and its corresponding measurements. These errorsincluded alignment errors and measurement errors underwater. Althoughthe frame mounting all the elements was rigid and aligned, the internalalignment of the light source and of the detectors may not have beenaligned perfectly along one axis. As a result, the profile measurementsof light along the z-axis and the along the xy-plane might be slightlyskewed. Another factor is the water turbidity. An accurate calculationof the water turbidity in a survey site is very challenging. Therefore,for more accurate distance detection algorithms, water turbidity shouldbe taken into account. The focus of the current study emphasized 3-Dtranslational motion. Future work will be towards expanding the researchto characterize rotational motion.

The study can be also applied in other applications, such as underwateroptical communication and docking. Underwater optical communication canprovide rates of up to 10 Mbits over ranges of 100 m. Several studieshave investigated the use of omnidirectional sources and receivers inseafloor observatories as a wireless optical communication. Anotherapplication is underwater docking by using optical sensors. Currently,studies have shown that such an application is possible for dockingvessels as far as 10-15 m for turbid waters and 20-28 m in clear waters.

Conclusions

Experimental work in this study was performed in order to evaluate thefeasibility of a control design for underwater distance detection. Theexperiments included beam diagnostics, spectral analysis and intensitymeasurements using a 50 W light source. A light source was mounted on arigid frame to the wall in the tow tank and a light detector was placedunderwater connected to a tow carriage that can move on rails along thetank. The study shows that a 500-550 nm band-pass filter will allow theobservation of a light field from a single source as a 2-D Gaussian beampattern. In the current configuration, the FWHM radius expands from 0.3m to 0.4 m as the array moves away from light from 4.5 m to a distanceof 8.5 m. During the empirical measurements in the UNH Tow Tank depth,alignment errors and measurement errors underwater were identified thatcan limit the performance of the distance detection algorithms.

Optical Detector Array Design for Navigational Feedback Between UUVs

Designs for an optical sensor detector array for use in autonomouscontrol of Unmanned Underwater Vehicles (UUVs), or between UUVs anddocking station, are demonstrated in this chapter. Here, various opticaldetector arrays are designed for the purpose of determining anddistinguishing relative 5 degree-of-freedom (DOF) motion between UUVs:3-DOF translation and 2-DOF rotation (pitch and yaw). In this chapter, anumerically based simulator is developed in order to evaluate varyingdetector array designs. The simulator includes a single light source asa guiding beacon for a variety of UUV motion types. The output images ofthe light field intersecting the detector array are calculated based ondetector hardware characteristics, the optical properties of water, andexpected noise sources. Using the simulator, the performance of planarand curved detector array designs (of varying size arrays) areanalytically compared and evaluated. Output images are validated usingempirical in situ measurements. Results show that the optical detectorarray is able to distinguish relative 5-DOF motion with respect to thesimulator light source. Furthermore, tests confirm that the proposeddetector array design is able to distinguish positional changes of 0.2 mand rotational changes of 10° within 4 m-8 m range in x-axis based ongiven output images.

Introduction

The underwater optical communication methods reported in the literatureare shown to be able to measure only up to 3-DOF, as opposed to theUUV's full maneuvering capabilities in all 6-DOF. Multiple DOF motion isnecessary to determine the relative orientation between two or more UUVsor between a UUV and a docking platform. Therefore, the design of anoptical detector array for such an application becomes crucial. Thischapter compares planar and curved array designs for underwater opticaldetection between UUVs or between a UUV and a docking station. Thecomparison between the two types of arrays is conducted using asimulator that models a single-beam light field pattern for a variety ofmotion types (i.e., 3-DOF translation and 2-DOF rotation). In addition,the number of elements in the array and the possible noise sources fromexperimental hardware and the environment are also taken into account.The results from the simulator are validated using in situ measurementsconducted in underwater facilities at the Jere E. Chase OceanEngineering Laboratory. The results of this study are to be used for thedesign of an optical detector unit for UUVs and the development oftranslational and rotational detection and control algorithms.

The performance criteria for an optical detector array design suitablefor underwater communication between UUVs can be judged by twocharacteristics. The first is the ability of the detector array toprovide a unique signature, that is, a sampled image that represents agiven location and orientation of a UUV with respect to a transmitter(i.e., light source). The second characteristic is the minimum number ofrequired optical detector components. This characteristic is derivedfrom the fact that a UUV should have a timely response to fast changesof the UUV's dynamics. (A smaller number of detectors would simplify thehardware design and reduce processing time. A unique signature, an imagefootprint from the optical detectors, would enable a UUV to receive thenecessary feedback to help the on-board control system to determineappropriate control commands to maintain a specified/desired orientationwith respect to and distance from a beacon (or any other object ofinterest).

Optical Design Considerations

The idea behind an optical detector array is such that as this array,which is mounted on a UUV, comes in contact with (without loss ofgenerality) a guiding beam, the light field is sampled and a signatureof the light beam can be obtained. Here, the light source represents aguide that is mounted on a leader UUV or on a docking station. In thisstudy, a single light source is used as the guiding beam for thedetector array. The light field generated from the light source isapproximated as a Gaussian beam at a given solid angle. For large arrays(i.e., arrays with several individual detectors), the light signaturecan be further represented as an image.

The design considerations for an optical detector array can becategorized as environmental and hardware-related. In this research, theprimary hardware for such a module consists of optoelectronic arraycomponents (e.g. photodiodes). These components are framed in a specificconfiguration and are mounted to an appropriate area on a UUV.

A planar array is an array of optical detectors that are mounted on aflat, 2-dimensional frame. Although the optical detectors can be placedin any configuration, a traditional equidistant design is assumed(without loss of generality) for the sake of simplicity. The detector,furthermore, is assumed to be square, having an equal number of verticaland horizontal elements (FIG. 3A(a)). The planar array simplifies thedesign and the resulting light signature, which is a cross-sectional(and possibly rotated) view of and within the light field. A curvedarray is an array of optical detectors that are mounted on either aspherical or parabolic frame. The geometry of the frame (curvature andoblateness) provides a larger range of incidence angles between thedetectors and the light field. In this study, all elements of the curvedarray are equidistant in a plane projection and located at a fixeddistance from the geometric center of the frame (FIG. 3A(b)). FIG. 3A isa schematic illustration of array designs used in the simulator: (a)Planar array and (b) Curved array.

Environmental Considerations

The light source in this study is assumed to be a point source with peakradiance L₀(r=0,ρ=0,Δλ) [W/m²·sr·nm] for a given detector with a fixedaperture area and a spectral range of Δλ. Using a cylindrical coordinatesystem, the axial distance from the light source to the optical elementalong the beam axis is defined as r and the radial distance from thebeam axis is defined as ρ. Assuming that light is not absorbed orscattered by the water medium, radiance collected by a detector isinversely proportional to the square of the distance to the source. Thelocation for half the peak intensity from the light source, Δr_(half),along the beam axis is assumed to be relatively small. The radiance fromthe light source according to the inverse-square law can be defined as:

$\begin{matrix}{{L_{obs}( {r,0,{\Delta \; \lambda}} )} = {{L_{0}( {0,0,{\Delta \; \lambda}} )} \cdot ( \frac{\Delta \; r_{half}}{r} )^{2}}} & (3.1)\end{matrix}$

Alternatively, the radiance change from one location, r₁, to a secondlocation, r₂, along the beam axis can be expressed using:

$\begin{matrix}{{L_{obs}( {r_{1},0,{\Delta \; \lambda}} )} = {{L_{obs}( {r_{2},0,{\Delta \; \lambda}} )} \cdot ( \frac{r_{2}}{r_{1}} )^{2}}} & (3.2)\end{matrix}$

The beam pattern produced from the intersection of a Gaussian beam lightfield with a plane that is perpendicular to the transmission directioncan be described using a Gaussian function. Traditionally, the beampattern is described using length terms with the peak intensity value atthe intersection point of the beam axis with the plane (ρ=0):

$\begin{matrix}{{L_{obs}( {r,\rho,{\Delta \; \lambda}} )} \approx {{L_{0}( {0,0,{\Delta \; \lambda}} )} \cdot ( \frac{\Delta \; r_{half}}{r} )^{2} \cdot ^{({- \frac{2\rho^{2}}{W^{2}{(r)}}})}}} & (3.3)\end{matrix}$

where, W(r) is the radial distance of the beam width on the plane at abeam intensity of 1/e² of the peak value at a distance r from the lightsource.

For this study, a description of the beam pattern angular terms wasapplied with a relationship: ρ=r·tan(η_(b)), where η_(b) is the anglebetween the beam axis and the light ray reaching the detector. Inaddition, the RMS width of intensity distribution, which is half of thebeam width, σ_(ρ)=0.5 W(r), was also converted to an angularrelationship:

$\sigma_{\eta} = {{\tan^{- 1}( \frac{\sigma_{\rho}}{r} )}.}$

Using a small-angle approximation, the exponent term can be defined as:

$\begin{matrix}{^{({- \frac{2\rho^{2}}{W^{2}{(r)}}})} = {^{({- \frac{r^{2}\tan^{2}\eta_{b}}{2r^{2}\tan^{2}\; \sigma_{\eta_{b}}}})} \cong ^{(\frac{- \eta_{b}^{2}}{2\sigma_{\eta_{b}}^{2}})}}} & (3.4)\end{matrix}$

Light in water is also attenuated by absorption and scattering.Environmental background noise, denoted by L_(b), from scattering oflight in the water column may occur. This attenuation can be describedusing Beer's law, which states that radiance decreases exponentiallythrough the medium as a function of distance, r, from the source and thediffuse attenuation coefficient, K(Δλ). The attenuated radiance at eachdetector is:

$\begin{matrix}{{L_{att}( {r,\eta,{\Delta \; \lambda}} )} = {{( {{L_{obs}( {r,\eta,{\Delta \; \lambda}} )} - L_{b}} ) \cdot ^{(\frac{{- 2} \cdot {K{({\Delta \; \lambda})}} \cdot r}{2 - \eta^{2}})}} + L_{b}}} & (3.5)\end{matrix}$

The environmental background noise caused by interaction between thelight beam and the water medium has been previously modeled. Thesestudies that have investigated the interaction of light beams throughturbulent medium approximate the background noise using a blurringfunction applied to the light beam. In this study, the background noiseis modeled using a Hanning window:

$\begin{matrix}{{h(n)} = {0.5( {1 - {\cos ( \frac{2\; \pi \; n}{N_{w} - 1} )}} )}} & (3.6)\end{matrix}$

where, N_(w), denotes the size of the Hanning window and n is the samplenumber in the window, i.e. 0≦n≦N_(w)−1. The Hanning window is convolvedwith the output image generated by the optical elements.

Hardware Considerations

As light interacts with a detector element (e.g., photodiode) in thearray, photons from the light are absorbed by the detector and currentis generated. The current is then manipulated by the signal conditioningcircuitry into a digital signal using an analog-to-digital convertor(ADC). The electrical signal measured by the detector is dependent onthe intensity (i.e., the optical power) of the light beam and on thedetector's responsivity (i.e., the electrical output of a detector for agiven optical input). Also, noise sources produced in the hardware canmake it difficult to extract useful information from the signal. Thequality of the detector is characterized by the sensitivity thatspecifies the minimum intensity value that can be detected. The keyhardware noise sources are: signal shot noise, σ_(s), background shotnoise, σ_(b), dark-current shot noise, σ_(dc), Johnson noise, σ_(j),amplifier noise, σ_(j), and ADC-generated quantization noise, σ_(q). Allsources of hardware noise are assumed to be mutually independent.Furthermore, it is assumed that all noise can be approximated asGaussian with corresponding values of standard deviation. Accordingly,these noise sources may be combined as a root sum of squares andrepresented with a net noise current:

$\begin{matrix}{\sigma_{n} = \sqrt{\sigma_{s}^{2} + \sigma_{b}^{2} + \sigma_{d\; c}^{2} + \sigma_{j}^{2} + \sigma_{q}^{2}}} & (3.7)\end{matrix}$

In addition to the electro-optical characteristics of the arraycomponent, the geometrical design of the array also affects the receivedintensity of the light signal. The incidence angle, θ, of the light rayreduces the level of radiance measured by the detector according toLambert's cosine law:

L _(θ)(r,η,Δλ)=L _(n)(r,η,Δλ)·cos(θ),  (3.8)

L_(θ)(r,η,Δλ)=L₀(r,0,Δλ)cos θ where L_(n) is the radiance at the surfacenormal.

The Simulator

Based on the hardware and environmental considerations, a simulator (ananalytical test bed) is developed. The goal of the simulator is toanalyze varying array designs for UUV optical detection of relativetranslation and rotation with respect to a reference coordinate frame.The criteria in evaluating the effectiveness of a detector array designincludes: 1) determining the minimum number of detector elementsrequired for robust UUV position and attitude determination and 2)verifying that the detector is able to acquire a unique signature foreach UUV position/orientation combination with respect to the givenlight source.

The simulator calculates light intensities at the individual opticalelements based on the relative geometry between the light source and thedetector. The simulator also takes into account the environmental andhardware effects described in the previous section. The effectiveoperational distance for underwater communication is dependent on waterclarity. Although a broad spectral range of light (400 to 700 nm) can beused for optical communication, the radiation calculation in thesimulator uses a narrower spectral range (between 500 to 550 nm),providing maximum transmittance in clear to moderately clear waters.Based on empirical measurements using a 400 W metal halide lamp and acommercial grade Mounted Silicon Photodiode photodetector, a maximumoperational distance of up to 20 m is assumed for extremely clearwaters, which represents open ocean conditions (K=0.05 m⁻¹), and up to 8m for moderately clear waters, which represents tropical coastal waters(K=0.1 m⁻¹). Although the simulator can provide results for largerangles, pitch and roll angles are limited to within 20°. This constraintis based on the assumption that most UUVs are built to be stable abouttheir pitch and roll axes of rotation.

Reference Frame

In the simulator, an Earth-fixed reference frame is assumed, where alight source is centered at the origin (0,0,0). Several coordinates areidentified in the x-y-z coordinate frame with respect to the UUV centerof mass (COM). Several attitude orientations are also identified withrespect to the Earth-fixed reference frame and defined by angles φ, θ,and ψ for roll, pitch, and yaw, respectively. In order to ensureappropriate sensor feedback for adequate control performance, thedetector array should be able to detect a unique light signal (pattern)for each combination of coordinate position and attitude orientation.Furthermore, this detection should be accurate to within 0.2 m of thetrue COM coordinate position and within 10° of the true attitudeorientation within 4 m-8 m range in x-axis.

The array geometry is chosen based upon the dimensions of the UUV. TheUUV in this study is assumed to be a rigid body of box-type shape with awidth (starboard to port) and height (top to bottom) of 0.4 m and alength (from bow to stern) of 0.8 m, the size of a genericobservation-class ROV used as a test platform. Accordingly, the widthand height of the detector array are 0.4 m×0.4 m for both planar andcurved array designs. The adapted coordinate axes convention is that ofthe Tait-Bryan angles. Here, the x-axis points toward the bow and they-axis towards starboard. The body-fixed z-axis points downward andcompletes the orthogonal triad. In this study, the follower is assumedto undergo rotation about all three-axes, i.e., pitch, roll and yaw. Thecoordinates associated with the array detectors are multiplied with therotation matrices in order to be in the same reference system as theleader UUV.

Array Geometry

As previously mentioned, two array shapes are compared in this study:(1) a planar array and (2) a curved array. The geometry of both arraysis defined in this section.

In the planar detector array, the detectors are defined relative to theUUV COM with respect to the local (body) coordinate frame. The centerand the four corners of the planar array frame are defined as follows:

$\begin{matrix}{{Arr}_{center} = ( {{{COM}_{x} + \frac{l}{2}},{COM}_{y},{COM}_{z}} )} & ( {3.9a} ) \\{{Arr}_{{\min {(y)}},{\max {(y)}}} = {{COM}_{y} \pm \frac{w}{2}}} & ( {3.9b} ) \\{{Arr}_{{\min {(z)}},{\max {(z)}}} = {{COM}_{z} \pm \frac{h}{2}}} & ( {3.9c} )\end{matrix}$

where COM_(x), COM_(y) and COM_(z) respectively define the x, y and zcoordinates of the follower COM, l is the length of the UUV, and w and hdenote the width and the height of the vehicle, respectively. Thelateral and vertical spacing (denoted as p_(y) and p_(z),) between theindividual detectors on the array can be expressed as:

$\begin{matrix}{p_{y} = \frac{w}{N - 1}} & ( {3.10a} ) \\{p_{z} = \frac{h}{N - 1}} & ( {3.10b} )\end{matrix}$

It is assumed that the detector array is an N×N square where N is thenumber of optical elements. That is, the number of detectors in the rowsand columns of the array are the same. Accordingly, the detector spacingis also the same (i.e. p_(y)=p_(z)). It is important to note that for acurved array, p_(y) and p_(z) are projected detector spacing.

A hemispherical shape is used for the curved array. The number ofdetectors in the curved array is initially defined based on the N×Nplanar array design. Then, if the detectors are projected onto thehemispheric surface, as in FIG. 3.1(b), with a fixed radius r:

x _(ij)=√{square root over (r ² −y _(ij) ² −z _(ij) ²)}  (3.11)

where x_(ij) is the position of the detector element on the x-axis andy_(ij) and z_(ij) are the coordinates of the array that is projectedonto the bow of the follower UUV. i and j are the indices that representthe row and column number of the array. In this study, the radius, r, ofthe hemisphere (of the curved array) is 0.32 m and is defined from itsfocal point, F, which is the center of the hemisphere:

F _(x)=COM_(x) +l/2  (3.12a)

F _(y)=COM_(y)  (3.12b)

F _(z)=COM_(z)  (3.12c)

The main difference between the planar and curved array designs is thatall of the optical elements in the planar array are oriented in the samedirection, while the detectors in the curved array are normal to thesurface of the array frame and thus allow a larger range of incidenceangles.

Radiometry

The construction of a realistic light field (as measured by the arraydetectors) is based on the radiometric and hardware considerations foreach detector. The radiometric calculations are based on the distance(i.e., inverse square law and Beer's law) and orientation (Lambert'scosine law) of each detector with respect to the light source. Using thedetector's characteristics and the associated electronics, theartificially created incident light is numerically converted into adigital signal. For the array simulator in this study, thespecifications of two types of photodiodes are used as reference(Thorlabs SM05PD1A, Thorlabs SM05PD2A). The resulting electronic signalis represented as a 10-bit (0-1023) sensor output value (thus,introducing quantization error). Environmental background noise isartificially added to the signal using a Hanning window of sizeN_(w)=11. Also, a random net noise current of σ_(n)=10⁻⁶ is added to theelectronic signal. The final digital signal is used to generate an imagepattern which, in turn, is to be used by the array detectors to identifythe position and the orientation of the UUV.

Results

Simulator Results

The success of the simulator described in this study relies on theability of the array to provide a unique image for every UUVposition/orientation combination. In order to process the simulatoroutput images more efficiently, the output data is reduced to a few keyimage parameters, allowing for a multi-parameter comparison. Thesechosen few parameters describe the beam pattern and allow the use ofsimple algorithms that do not require significant computational effort.One such algorithm is the Spectral Angle Mapper (SAM), which is the dotproduct between sets of key parameters extracted from two images thatare represented as vectors, U(u₁, u₂, . . . u_(np)) and V(v₁, v₂, . . .v_(np)):

$\begin{matrix}{\alpha = {\cos^{- 1}( \frac{\overset{arrow}{U_{t}} \cdot \overset{arrow}{V_{t}}}{{\overset{arrow}{U_{t}}} \cdot {\overset{arrow}{V_{t}}}} )}} & (3.13)\end{matrix}$

The calculated angle between the two vectors, i.e. SAM angle α, is thenumerical resemblance between the images. Two very similar images resultin an angle value close to 0°, whereas two very different images resultin an angle close to 90°. The SAM angle provides a good performanceevaluation indicator to the different types of array detector geometriestested using a single-value parameter.

Although the UUV is a six DOF system, it is assumed that it is notpossible to achieve relative roll angle detection (because of axialsymmetry about the body x-axis). Thus, five parameters are provided tothe simulator as input: translation along all three coordinate axes,rotation of the pitch angle, θ, and rotation of the yaw angle, ψ.Accordingly, the image output of the simulator is analyzed using fiveparameters that can be related to input parameters (FIG. 3B): the peaklight intensity value, I, the corresponding location of the horizontaldetector, j, and vertical detector, k, at peak intensity, the locationof the skewness of the horizontal intensity profile gradient, Sk_(h),and skewness of the vertical intensity profile gradient, Sk_(v). Thepeak value is normalized with respect to a given maximum detectableintensity (0.0<I<1.0). The locations of the horizontal and verticaldetectors are defined with respect to the central detector (j=(N+1)/2,k=(N+1)/2). Based on the location of the peak intensity, the slopes ofthe horizontal and vertical intensity are calculated. The slope of theprofile is used rather than the profile itself as the slope alsoprovides the directionality of the beam profile (i.e., negative orpositive) in addition to the asymmetry of the profile. The images andthe corresponding parameters for the planar and the curved array of size21×21 for a given coordinate location and yaw rotation are shown in FIG.3B and FIG. 3C, respectively.

FIG. 3B shows key image parameters and intensity profiles for a planararray detector unit with hardware and environmental background noise:(top left) Output image from the simulator, (top right) Horizontalprofile, (bottom left) Vertical profile, (bottom right) Input valuesused to generate output image and key parameters describing outputimage.

FIG. 3C shows key image parameters and intensity profiles for a curvedarray detector unit with hardware and environmental background noise:(top left) Output image from the simulator, (top right) Horizontalprofile, (bottom left) Vertical profile, (bottom right) Input valuesused to generate output image and key parameters describing outputimage.

Detector Array Comparison

As a first step for the selection of the array design, the geometry ofthe detector array is evaluated. A performance evaluation between planarand curved arrays is conducted, where each detector array contains a21×21 grid of detector elements with a detector spacing of 0.02 m. Bothdetector arrays are evaluated for their ability to detect changes inposition and orientation, i.e., changes in SAM angle, α. Changes inposition are evaluated as the UUV translates along the y-axis from agiven origin (0 m) to an offset of 0.9 m in 0.03 m increments.Similarly, changes in orientation are evaluated by rotating the UUVabout the z-axis, yaw rotation, from its initial reference (0°) to 30°in increments of 1°. FIG. 3D represents the resemblance results toidentify UUV positional and attitude changes based on measured signals(images) collected by the detector array at 4 m. The comparative resultsfor changes in position using the SAM algorithm show similar performancebetween the two array geometries, where the curved array performsslightly better (2°) at shifts greater than 0.6 m. However, aninvestigation of the results for changes in orientation reveals that thecurved array is more sensitive to changes in orientation than the planararray. The SAM angle results for the curved array show changes of 12° at5° yaw rotations and changes of 22° at 10° rotations, whereas theresults for the planar array show changes in SAM angle of 5° at 5° yawrotations and 11° at 10° rotations. Based on these results, it isdeduced that the curved array geometry is more suitable fordistinguishing changes in position and, especially, orientation of a UUVplatform with respect to a reference light beacon. FIG. 3D illustratescomparative resemblance results (SAM angles) for 21×21 element curvedand planar array (at x=4 m) as a function of: (a) lateral translation,(b) yaw rotation.

After the geometry of the detector array is defined, relationshipsbetween the ability to distinguish changes in position and orientationfrom the output images and the number of elements in the curved detectorarray are evaluated. The comparisons include different array sizes,ranging from a 3×3 size array up to a 101×101 size array at distancesranging from 4 m to 8 m to the light source. The comparative results at4 m (FIG. 3E) show that changes in positional and rotational shifts canbe detected by an array with the size of at least 5×5 optical elementswith detector spacing of 0.1 m. Based on a threshold of a 15° SAM angle,a smaller array would fail to detect translational shifts smaller than0.2 m or rotational changes smaller than 10°. It should also be notedthat no significant changes in detection capability are observed forarray sizes greater than 7×7 with a detector spacing of 0.067 m.Although the ability of the curved array to distinguish between theimages decreases as the operational distance increases, the SAMalgorithm results for 5×5 array at 8 m are still above 10° for a 10° yawrotation and above 6° for 0.2 m translation.

FIG. 3E illustrates comparative resemblance results (i.e., SAM angle)with respect to varying array sizes (incorporating environmental andbackground noise): (a) SAM angle with respect to lateral motion (b) SAMangle with respect to angular rotation.

FIG. 3F illustrates comparative resemblance results (i.e., SAM angle)with respect to operational distance (incorporating environmental andbackground noise): (a-c) lateral shift, (d-f) yaw rotation—(a,d) 3×3array (b,e) 5×5 array (c,f) 101×101 array with spacing of 0.2 m, 0.1 mand 0.004 m, respectively.

Experimental Confirmation

In addition to the analytical study presented in this chapter,experimental validations are conducted at the wave and tow tank. Theunderwater experiments compare the simulator outputs to that ofempirical measurements. This comparison validates the optical model usedin the simulator (i.e., Gaussian beam profile) and confirms theenvironmental physical properties that contribute to the light field asreceived by the detector array. The light source in this study is a 400W underwater halogen lamp (contained in a waterproof fixture). Profilesof light intensity data (radiance measurements) were collected via aspectrometer such that the measurements are perpendicular to that of theillumination axis. The profiles are collected at distances ranging from4 m to 8 m at 1 m increments and with lateral shifts from theillumination axis up to 1 m away from the axis at 0.1 m increments.

The profiles from empirical measurements are compared to profilesproduced from simulator output images calculated for the same distanceand orientation conditions (FIG. 3G). The measured profiles confirm thatthe light field calculations for the simulations are valid. Although thebackground noise in the simulated models is overestimated, thecorrelation, R², between the two profiles is between 0.95-0.99 fordistances from 4-8 m. FIG. 3G illustrates comparison of Experimental andSimulation results (a) 4 m (b) 5 m (c) 6 m (d) 7 m (e) 8 m.

Discussion

The results of this study show that the detector array simulator is auseful and reliable tool for array design in optical communicationbetween UUVs or between a UUV and a docking station. The simulator has amodular design to allow for the addition and changing of hardware andenvironmental parameters. Although the simulator can evaluate otherarray geometries with a variety of sizes, only two traditional shapesare considered. The simulator results show that a curved array with aminimum array size of 5×5 elements is sufficient for distinguishingpositional changes of 0.2 m and rotational changes of 10°. For thedistinction of smaller changes, a larger array size is required.

A follower UUV is assumed to have five DOF maneuverability with respectto a given light source: three DOF translations (i.e., translationsalong the x, y, and z axes) and two DOF rotations (yaw and pitch).Because the transmitter unit in the presented configuration has only onelight source with a Gaussian spatial intensity distribution, it is notpossible to decouple roll changes (rotation about the body-fixed x-axis)from either pitch or yaw. This is due to the axial symmetry of the lightbeam. The use of multiple light sources or a light source with a uniqueintensity distribution may enable roll rotation sensing.

It is important to note that the simulator assumes that the water columnis uniform with systematic background noise. As a result, the outputimages of the light field intersecting with the detector arrayresemblance a Gaussian beam pattern. However, disturbances in the medium(e.g., sediment plume) may cause the beam pattern to be distorted. Thispoint should be taken into account in the development of controlalgorithms for UUV navigation. Otherwise, the control algorithms maymisinterpret the acquired image and direct the follower UUV away fromthe guiding beam. The simulator results show that detector noise doesnot contribute significantly to the image output. Other detectors with alarger noise level may contribute more to output images.

An alternative hardware component that was considered instead ofphotodetectors was a camera array. The potential benefits using COTScameras (CCD or CMOS) is to provide additional spatial information thatcan potentially enhance the performance of pose detection algorithms.However, one of the requirements for an autonomous system is the abilityto process the sensor's input and execute the pose detection algorithmsfast enough to respond to changes in the UUV's dynamics (i.e., detectionof the leader UUV and a response by the follower UUV). It seems that acamera array that performs image extraction and processing proceduresmay not be sufficiently fast for the UUV interaction. With that said,the camera array option will be considered for future work that allowsslower update rates of pose detection algorithms.

Conclusions

In this chapter, a detector array simulator was developed to evaluatedifferent geometrical structures of optical arrays of varying sizes forunderwater position/orientation detection between UUVs or between a UUVand a docking station. Criteria for an array design suitable inunderwater communication between UUVs was based on: 1) the ability ofthe array to distinguish changes in position and orientation of a UUVwith respect to a given light source and 2) the minimum number ofoptical detector components that would simplify the hardware design andreduce processing time. The simulator calculated a light field generatedfrom a single light source passing through the water column, taking intoaccount attenuation and scattering. Based on the optoelectroniccharacteristics of the detectors (including noise) and the array design,an output image of the light field intersecting the detector array wasproduced.

Two array designs, i.e. planar and curved, were evaluated based on theirability to distinguish changes in position and orientation between theUUV and a guiding light source. Because of the beam pattern symmetry(generated from a single light source), it is possible to detect 5-DOFUUV motion, i.e. translations along x, y and z-axes, as well as pitchand yaw rotations, and not full 6-DOF motion. The input data in thesimulator evaluation included the relative geometry between the lightsource and the optical array. SAM algorithm evaluated the detector arraydesign and was able to distinguish translational changes within anoperational range between 4 m to 8 m with accuracy of 0.2 m androtational shifts within 10° using key output image parameter values.Using a 21×21 array with detector spacing of 0.02 m, it was determinedthat a curved array design is more sensitive to rotational changes thana planar array, whereas both array geometries performed similarly fortranslational shifts. After the geometry of the detector array wasdefined, the minimum number of element in the detector array wasdetermined. The simulator results showed that an array of at least 5×5detector elements with 0.1 m detector spacing was needed to distinguishchanges in five DOF. The results were also validated using in situexperimental measurements.

Pose Detection and Control Algorithms for Dynamic Positioning of UUVsVia an Optical Sensor Feedback System

The use of an optical feedback system for pose detection of UnmannedUnderwater Vehicles (UUVs) for the purpose of UUV dynamic positioning isinvestigated in this chapter. The optical system is comprised of acurved optical detector array (on board the UUV) of hemisphericalgeometry that is used to detect the relative pose between an externallight source and the UUV. This pose detection is accomplished in twoways: via Spectral Angle Mapper (SAM) algorithm and via image momentvariants. These two methods are also compared to a traditional imageprocessing algorithm, phase correlation and log-polar transform. In thischapter, analytical simulations are conducted to test the efficacy offeedback controllers (PID, Sliding Mode Control) using the opticalfeedback system and a previously developed numerical simulator. Theresulting dynamic positioning and control performance of a UUV isobserved in two simulated control scenarios: 1) a static-dynamic(regulation control) system in which the UUV autonomously positionsitself via 4 degrees-of-freedom (DOF) (translational control in additionto yaw/heading control) with respect to a fixed external light sourceand 2) a dynamic-dynamic (tracking control) system where one UUV tracksanother independent UUV (via translational control in 3 DOF). In thesesimulations, the numerical simulator takes into account environmentalconditions (water turbidity and background noise) and hardwarecharacteristics (hardware noise and quantization). Simulation resultsshow proof of concept for this optical-based feedback control system forboth the static-dynamic and dynamic-dynamic cases, the UUV being able toregulate/track its desired position(s) to within a reasonable level ofaccuracy.

Introduction

As a first step to investigate positioning and coordinated formation ofUUVs using optical communication, a detector array interface, i.e. anumerical simulator was designed. An optical array for UUVs was designedbased on theoretical models of a point source light field and a range ofoceanic conditions (e.g. diffuse attenuation coefficients). In thisstudy, a curved optical detector array design of hemispherical geometrywith radius of 0.55 m is used to decouple UUV translation fromorientation changes using sensor detection measurements of an externallight source. Array sizes of 21×21 and 5×5 grids of detection sensorsare investigated in order to observe comparative pose detectionperformance capabilities. The detection algorithms are developed basedon two different types of underwater applications. The first one is astatic-dynamic (regulation control) system, e.g. UUV approaching adocking station or a data-transfer hub, in which a guiding light source(and, therefore, its optical illumination axis) is static. Here, the UUVis controlled to arrive at and maintain a desired position andorientation with respect to the light source. The second underwaterapplication is a dynamic-dynamic (tracking control) system, e.g.leader-follower UUV tracking system. In this case, one leader UUV isequipped with a light source with a Gaussian intensity profile mountedto its crest and a follower UUV is guided with the aid of a lightdetector array mounted at its bow. Two types of pose detectionalgorithms (Spectral Angle Mapper (SAM) and image moment invariants) arecompared to a more traditional image processing approach (phasecorrelation and log-polar transform). Performance criteria of the threealgorithms include positional accuracy, processing speed, and dependenceon the environmental characteristics. In this chapter,Proportional-Integral-Derivative (PID) and Sliding Mode Controllers(SMC) are implemented for a variety of static-dynamic anddynamic-dynamic UUV scenarios in order to evaluate the optical-basedfeedback control system performance under varying conditions.

Pose Detection Algorithms

Phase Correlation and Log-Polar Transform

Phase correlation and log-polar transform approach to pose detection isable to take into account images in 4-degrees-of-freedom (DOF) (i.e.,rotation, scale and translation along two axes). The phase correlationalgorithm uses Fourier Shift Theorem to detect the translated images. Itis given that two images, represented as f₁ and f₂, observe the sametarget source acquired at different locations with relativetranslations, dx and dy, with respect to each other. Then, at the samerelative orientation, the relationship between the two images can bedescribed as:

f ₂(x,y)=f ₁(x−dx,y−dy)  (4.1)

The corresponding relationship of the Fourier transforms for theseimages, F₁(ω_(x), ω_(y)) and F₂(ω_(x), ω_(y)), is given by:

F ₂(ω_(x),ω_(y))=e ^((ω) ^(x) ^(dx+ω) ^(y) ^(dy)) F₁(ω_(x),ω_(y))  (4.2)

The magnitudes of F₁(ω_(x), ω_(y)) and F₂(ω_(x), ω_(y)) are comparableto each other if the relative translations, dx and dy, are, in turn,comparatively small with respect to the image size, whereas the phasedifference between the two images is directly related to theirtranslation. This phase difference is equivalent to the phase of thecross-power spectrum:

$\begin{matrix}{^{{\omega_{x}{dx}} + {\omega_{y}{dy}}} = \frac{{F_{1}( {\omega_{x},\omega_{y}} )}{F_{2}^{*}( {\omega_{x},\omega_{y}} )}}{{{F_{1}( {\omega_{x},\omega_{y}} )}{F_{2}^{*}( {\omega_{x},\omega_{y}} )}}}} & (4.3)\end{matrix}$

where F₂*denotes the complex conjugate of F₂.

The relative translation values are derived by calculating the inverseFourier Transform in (4.3). The location of the resulting peakcorresponds to the translation of dx and dy, respectively, such that

(dx,dy)=max(F{e ^(ω) ^(x) ^(dx+ω) ^(y) ^(dy)})  (4.4)

The rotation and scale between two images is calculated using thelog-polar transform. Here, both images are first translated from aCartesian domain (x, y), to a log-polar domain (log(φ, θ) using thefollowing transformation:

$\begin{matrix}{{\log (\rho)} = {\log( \sqrt{( {x - x_{c}} )^{2} + ( {y - y_{c}} )^{2}} )}} & (4.5) \\{\theta = {\tan^{- 1}( \frac{y - y_{c}}{x - x_{c}} )}} & (4.6)\end{matrix}$

where ρ is the radial distance from the center of the image, (x_(c),y_(c)) and θ is the corresponding angle (FIG. 4A). FIG. 4A showstransformation of an image from Cartesian space to polar space [90].Cartesian space (left). Polar space (right).

After the transformation of the images to a log-polar domain, the phasecorrelation algorithm described in (4.3) is applied to detect relativerotation and scale between the two images.

A reference image is calculated and obtained from the simulator and isdesignated as f₁. The pose parameters for the reference image are a setof five pre-defined x-axis offsets (e.g., 4 m to 8 m with 1 mincrements) with respect to the leader's beacon. All other 5-DOF posegeometries (translation with respect to the y and z axes; roll, pitchand yaw rotations) are kept the same as the leader UUV. Theinstantaneous image of the follower UUV in motion, f₂, is calculatedusing the simulator for the pose evaluation path. The results from thephase correlation and log-polar transformation algorithms (i.e.,relative translation and rotation between the reference and theinstantaneous image) are converted to the local coordinate referencesystem showing the relative translations and rotations between theleader and that of the follower. The algorithms are evaluated in termsof their correlation to the parameters in the pose evaluation pathdatasets.

Spectral Angle Mapper (SAM)

Key parameters from the follower's detector array output image wereextracted to a vector of identifiers for each pose. Changes intranslation and orientation between the poses were monitored using a dotproduct between two identifier vectors of two poses using SAM describedin Chapter 3. In this chapter, the SAM algorithm relies on five mainimage parameters that include the skewness of both the row and column ofthe resulting intensity profile of the image pixel with the maximumintensity and the row and column numbers of the image pixel with themaximum intensity as demonstrated.

The SAM algorithm has been implemented for a planar detector array of21×21 elements. To quantify the amount of shift in x-axis, an offlinecalibration procedure that ranges from 4 m to 8 m at 1 m increments isperformed. The reference image is the output obtained when there is onlyx-offset, i.e. no translation in the y and z axes or yaw or pitchrotations. (Roll rotation is not considered, as it is not possible todetect roll changes from a single light source configuration.) The imageunder test is the output when there is a specific relative geometry,including all of 5-DOF motions (again, full 6-DOF motion less rollrotation), between the light source and the detector. Images areproduced for all possible poses of the follower UUV with respect to theleader UUV over a translation range ±0.3 m at 0.03 m increments and arotation range of ±30° at 3° increments. A vector of the five main imageparameters is extracted for each image. This reference dataset, which isa look-up table of vector identifiers, is used to calculate the pose ofthe instantaneous images. Instantaneous image vector identifiers arecompared to vector identifiers in the aforementioned look-up table. Posecandidates based on the location of image pixel (row and column numbers)with the maximum intensity are extracted from the look-up table. Theextracted pose candidates are then weighted with the weightingcoefficients to form a cost function using:

P _(i) =c ₁ |Sk _(x) −Sk _(xi) |+c ₂ |Sk _(y) −Sk _(yi) |+c₃|SAM−SAM_(i)|  (4.7)

where P_(i) is the cost function, c₁, c₂ and c₃ are the weightingcoefficients that are determined by trial and error, Sk_(xi), Sk_(yi)and SAM_(i) are the skewness among the row, skewness among the columnand SAM angle for the instantaneous pose, i. The resulting pose isdetermined based on the pose candidate that results in the minimum costfunction, P_(i). The algorithm is evaluated based on the accuracy of thetype of pose detection (i.e. relative translation and motion) betweenthe leader and the follower vehicle and the amount of shift intranslation and rotation. Pose detection algorithms using SAM approachwere explained in more detail.

Calculation of the Image Moment Invariants

The third approach used for pose detection in this study utilizes theimage moment invariants that are defined as the weighted sum of theintensity values of the array pixels, l_(i,j) with respect to thelocation of the peak intensity, P_(max)=(y_(o), z_(o)) [93]-[94]. Imagemoment invariants can be defined as:

$\begin{matrix}{M_{pq} = {\frac{1}{S}{\sum\limits_{i,j}\; {( {y_{i} - y_{o}} )^{p}( {z_{j} - z_{o}} )^{q}I_{i,j}}}}} & (4.8)\end{matrix}$

where S=Σ_(i,j)I_(i,j) and y_(i), z_(j) are the row and columncoordinates for a given detector in the array, respectively. p and qdenote the order of moments. Image moment invariants are calculated upto the second order (p=2, q=2) for maintaining a simple and efficientcalculation of the pose detection, i.e. M₀₀, M₁₀, M₁₁, . . . M₂₂. Thelocation of the pixel with the maximum intensity, P_(max), is calculatedat a sub-pixel accuracy. The relative translational and rotationalmotions between the UUVs in this approach are distinguished based onmoment invariant functions. The output of the image moment invariantsalgorithm for a specific pose is a 3×3 matrix where each element denotesinformation about the symmetry of the light intersected on the array.The pose detection algorithm utilizes this property of the image momentinvariants approach in a calibration procedure to determine and quantifythe pose.

UUV Modeling and Control

The UUV pose-based feedback control system is deemed as successful ifthe UUV in study is able to maintain relative pose to within an accuracyof ±0.1 m in translation (i.e. in each x, y and z-axes) and ±5° inrotation (yaw and pitch). These requirements are to accomplish dockingoperation of a UUV into a docking station. The PID and SMC areimplemented separately on a simulated UUV system under both theregulation (static-dynamic) and tracking (dynamic-dynamic) scenarios.

UUV control in this study is restricted to be decoupled. That is, theUUV is allowed to be controlled in 1-DOF at a time, (i.e., either asingle x, y, or z translation or pitch or yaw rotation). This is truefor both the leader and the follower UUVs. The leader UUV is assumed tobe controlled separately, say, with a user-controlled joystick(open-loop) whereas the follower UUV has PID and SMC implemented forautonomy.

For a dynamic-dynamic system under the assumption that the leader UUVhas a known path a priori, the follower UUV can use informationcollected by the curved detector array as feedback to determine theleader UUV's relative pose

η_(f)=η_(l)−η_(d)  (4.9)

where η_(f) is the follower pose, m is the leader pose determined by thefollower, and η_(d) is the desired relative pose, incorporating desiredrelative distance and attitude, between the leader and the followerUUVs. The control problem in this case can be evaluated as a trajectorycontrol problem as the leader is assumed to be remotely controlled togiven waypoints while the follower generates its own time-varyingtrajectory from the leader motion. For a static-dynamic system, theproblem can be considered as a set-point regulation problem.

Results

The image moments approach requires a calibration procedure in order todistinguish the motion type and quantify the degree of relativetranslational and rotational displacements. In the case of a singleguiding light beam, the calibration procedure is only conducted forpositive translation and rotation values because the image momentinvariants for positive translation and rotational motion aresymmetrical. (A sign difference exists only for negative motion.) For a21×21 sized light sensor detector array, image outputs from the arraysimulator include translation, ranging from 0 to 0.18 m in 0.02 mincrements in the y and z-axis direction and from 0° to 27° with 3°increments for pitch and yaw rotations. For a 5×5 sized array, thecalibration range for translation motion is from 0 to 0.14 m in 0.02 mincrements and from 0° to 27° with 3° increments for pitch and yawrotations. In the x-axis direction, the calibration range between theleader and the follower vehicle is from 2 m to 8 m at 1 m increments.The limiting factor of the calibration range is the sub-pixel accuracyalgorithm in both translational and rotational motion because thisalgorithm requires the intensity value of the neighbor pixel that isadjacent to the pixel with maximum intensity. Because of thislimitation, the pixel with the maximum intensity cannot be located at aborder of the array. The calibration procedure consists of two steps: 1)calibration for relative decoupled motion (i.e. the UUV being restrictedto a single 1-DOF motion at a time) and 2) calibration to detect motionwhen the UUV is translated in y-axis and rotated in yaw.

Static-Dynamic System

The pose detection algorithm is first based on distinguishing the typeof motion (i.e. translational from rotational). This is accomplished byevaluating numerical values of the image functions which provide adescriptive feature of a specific motion type, i.e. translational orrotational motion. After the type of motion is determined, the amount oftranslational or rotational displacement is quantified through anoff-line calibration procedure to linearly estimate the amount ofrelative shift in both translation (x, y and z-axis directions) androtation (pitch and yaw) between the leader and the follower vehicleswith respect to detected light levels.

In the static-dynamic system algorithm, it is assumed that the relativemotion between the fixed light source and the UUV consists of 5-DOFmotion (all 6-DOF except roll). In addition, pitch is not consideredwhen the UUV aligns itself with the external light source when the UUVis initially misaligned along 4 axes of translation and yaw. Thedetection and control strategy for the static-dynamic system is firstbased on quantifying the UUV z-axis motion and control to its desiredstate, z_(d). The second step is to distinguish between y-axistranslation and yaw motion. This second step is particularly complex asy-axis translation and yaw rotation both act on the same axis and candistort the detection algorithm and, therefore, degrade the accuracy ofthe results. The second calibration procedure is conducted todistinguish and quantify yaw rotation and y-axis translation when bothmotions are present.

In this section, two sets of results are presented. The first set ofresults shows the detection and control capability of the pose detectionalgorithms combined with SMC and PID controllers when there is only oneaxis of translation or rotation that is initially misaligned. For eachDOF, the UUV is directed to a desired state. The second set of resultsconsists of case studies in which all pose detection capabilities andmodels are utilized. In this case, 4 axes of translation/rotation aremisaligned (x, y, z and yaw) and the UUV has four desired states toreach. This is also conducted introducing a constant external current(disturbance). The control loop for all the simulations are run at arate of 10 Hz.

Single-DOF SMC, Motion with 21×21 Detector Array

In this scenario, a UUV is mounted with a curved detector arrayconsisting of 21×21 photodetector elements. A single stationary lightsource placed underwater acts as a guiding beacon to position the UUVwith respect to a given reference value. 1-DOF SMC control isestablished for each of the five axes (i.e. x, y, z, pitch and yaw)separately. It is assumed that there are no external disturbances in theenvironment. The simulation results in FIG. 4B show that with the posedetection algorithms and SMC, the UUV in study converges to the desiredreference point for each DOF. In addition, Table 4.1 shows the resultsfor decoupled translation and rotation, showing UUV initial conditions,reference (desired) position/orientation and final position/orientation.FIG. 4B shows independent DOF SMC results for a curved 21×21 array. (a)x-axis control (b) y-axis control (c) z-axis control (d) yaw control (e)pitch control.

TABLE 4.1 Initial, reference and final UUV positions and orientationsfor a 21 × 21 curved array for decoupled 5-DOF Control Initial position/Reference position/ Final position/ orientation orientation orientationx_(i) = 8 m x_(d) = 4 m x_(f) = 4.02 m y_(i) = −0.1 m y_(d) = 0.1 my_(f) = 0.09 m z_(i) = 0.1 m z_(d) = −0.1 m z_(f) = −0.09 m ψ_(i) = 10°ψ_(d) = −10° ψ_(f) = −10.05° θ_(i) = 10° θ_(d) = −5° θ_(f) = −5.03°

Single DOF SMC, Motion with 5×5 Detector Array

It is shown that a 5×5 curved detector array was sufficient todistinguish between the translational and rotational displacements. Inaddition, the construction of a 5×5 array is more cost-efficient thanthe construction of a 21×21 array. Therefore, pose detection and controlalgorithms are also developed for a 5×5 array. SMC is implemented in thesame way as applied in the case of the 21×21 array. The simulations areconducted without any external disturbance present. The pose detectionalgorithm and SMC work efficiently to dynamically position the UUV tothe vicinity of the desired reference values (FIG. 4C and Table 4.2)especially in the translational directions. In both cases, i.e. a 21×21and a 5×5 detector arrays, it is observed that small amplitudeoscillations exist for yaw and pitch control. As it is shown that posedetection and SMC for a 5×5 array demonstrate satisfactory controlperformance and that a more costly option of a 21×21 array is notnecessary, further simulations in this study are conducted solely with a5×5 detector array. FIG. 4C shows independent DOF control results withSMC for a curved 5×5 array. (a) x-axis control (b) y-axis control (c)z-axis control (d) yaw control (e) pitch control.

TABLE 4.2 Initial, reference and final UUV positions and orientationsfor a 5 × 5 curved array for decoupled 5-DOF Control Initial position/Reference position/ Final position/ orientation orientation orientationx_(i) = 8 m x_(d) = 5 m x_(f) = 4.99 m y_(i) = −0.1 m y_(d) = 0.1 my_(f) = 0.09 m z_(i) = −0.14 m z_(d) = 0.14 m z_(f) = 0.14 m ψ_(i) = 20°ψ_(d) = −20° ψ_(f) = −17.7° θ_(i) = −20° θ_(d) = 20° θ_(f) = 15.3

x-Axis PID Control with a 5×5 Detector Array

In addition to the SMC to dynamically position a UUV to a desireddistance and orientation, the effectiveness of a PID controller is alsoinvestigated. The PID controller is tested for the control oftranslation only in the x-axis direction and without any disturbancespresent. The selected PID controller gains are P=400, l=2 and D=300. ThePID controller does, in fact, eventually control the UUV to the desiredreference value (FIG. 4D). However, after a reasonable effort to tunecontrol gains, excessive overshoot still remains, more specifically a30% maximum percent overshoot. PID controller performance is acceptablefor translational control about the x-axis, but a potential overshoot isnot acceptable for y and z-axis translational control because it causesthe UUV to lose its line of sight with the external light source.Therefore, it is concluded that PID control is considered unsuitable forthis application. FIG. 4D shows a PID x-axis control for a 5×5 array.

Case Study: Dynamic Positioning with Multiple Concurrent Initial PoseErrors

After it is determined that a 5×5 array with the implementation of SMCis suitable for pose detection and control of a UUV, a dynamicpositioning case study is conducted. In contrary to the cases describedin the previous sections, the UUV is given 4-DOF “off-axis” initialconditions (i.e., concurrent non-zero errors in x, y, z and yaw). Thegoal is to dynamically position the UUV with respect to the fixed,single beam light source with a desired position and orientation. Thecontrol strategy in this case study is for the UUV to perform decoupledcontrol actions, one DOF at a time, until the desired reference value isfinally reached, where the decoupled control sequence is as follows: 1)z-axis control 2) yaw control 3) y-axis control 4) x-axis control. Theinitial, reference and final positions and orientations are listed inTable 4.3.

TABLE 4.3 Initial, reference and final UUV positions and orientationsfor a 5 × 5 curved array for decoupled 4-DOF control Initial position/Reference position/ Final position/ orientation orientation orientationx_(i) = 8 m x_(d) = 4 m x_(f) = 4.12 m y_(i) = 0.14 m y_(d) = 0 m y_(f)= −0.01 m z_(i) = 0.14 m z_(d) = 0 m z_(f) = 0.02 m ψ_(i) = −20° ψ_(d) =0° ψ_(f) = −0.1°

The results demonstrate that the UUV accomplishes the control task towithin reasonable accuracy (FIG. 4E). There is an offset of 0.02 m forthe z-axis. The calibration procedure for y and yaw detection is basedon the case where the UUV z-axis coordinate is perfectly aligned withthe corresponding z-axis coordinate of the external light source. Thus,the pose detection algorithm is robust enough to produce accurateestimates in the presence of relatively small steady-state errors. Inthe second step, yaw control is quite accurate with 0.1° error and thenthe control system switches to y-control mode, where it stops at t=80 swhen a satisfactory y-axis position is reached. In the final step, thex-axis coordinate is controlled with a steady-state error of 0.12 m. Itshould be noted that after y-axis control stops and x-axis controlbegins, there is a slight change in the UUV y-axis coordinate. This isdue to the steady-state error in yaw. FIG. 4E shows UUV docking casestudy using SMC for a 5×5 array. The UUV with four initial non-zero poseerrors is commanded to position itself with respect to a fixed lightsource. (a) x-axis control (b) y-axis control c) yaw control d) z-axiscontrol.

Case Study: Dynamic Positioning with Multiple Concurrent Initial PoseErrors in the Presence of Added Disturbances

In this scenario, the UUV initial conditions and the references are keptthe same as that in Table 4.3, but a constant current (externaldisturbance) of −0.03 m/s in the x-axis direction is introduced. It isobserved from the results (FIG. 4F) that z-axis control is notsignificantly affected by the added current but does have a notedsteady-state error. And, although yaw control is affected by thecurrent, it stabilizes at 8°. Due to the yaw steady-state offset, whenthe UUV moves along its body-fixed coordinate x-axis, the UUV also movesin the Earth-fixed coordinate y-axis. This can be observed from t=7 s.to t=82 s. In this period, there is only x-axis control. Then, thecontroller switches to y-axis control and it attempts to regulate to they-coordinate but does so with some chatter. As the UUV approaches tox-axis desired reference, the SMC switches between x and y-axis controlto the end of the simulation. FIG. 4F shows a UUV docking case studyusing SMC for a 5×5 array with a current of −0.03 m/s in x-axis. (a)x-axis control (b) y-axis control c) yaw control d) z-axis control.

Dynamic-Dynamic System

In addition to the static-dynamic scenario in which the optical-basedsensor system uses SMC to control a UUV with respect to a fixed lightsource, the capabilities of the dynamic-dynamic (tracking control)scenario is also investigated. In the dynamic-dynamic system, there aretwo moving UUVs, a leader UUV and a follower UUV. The leader UUV has alight source at its crest and is assumed to move independently, forexample, via remote operation. The follower UUV processes the sampledlight field coming from the leader UUV on the detector array andperforms the appropriate control action autonomously in order to trackthe leader. In this scenario the leader and the follower UUV initialconditions, reference values and the final positions are given in Table4.4.

TABLE 4.4 Initial, reference and final UUV positions and orientationsfor a 5 × 5 curved array for dynamic-dynamic control Initial LeaderReference Final Leader position position position Final Followerposition x_(il) = −8 m x_(d) = −8 m x_(fl) = −9.75 m x_(ff) = −1.5 my_(il) = 0 m y_(d) = 0 m y_(fl) = 0.10 m y_(ff) = 0.14 m z_(il) = 0 mz_(d) = 0 m z_(fl) = 0.11 m z_(ff) = 0.14 m

Initial evaluation of yaw control in a dynamic-dynamic scenario (notshown here) reveals that relatively small deviations in yaw cause lossof line-of-sight between the UUVs. Therefore, yaw control in adynamic-dynamic system is not pursued in this study. The controlled DOFbetween the UUVs are, instead, translation about all three coordinateaxes. SMC is implemented for the control of a UUV mounted with a 5×5detector array. In this simulated scenario, both the leader and followerUUV start with an 8 m offset in x-axis direction and no offset along they and z-axis directions. The control goal for the follower UUV is tomaintain these initial conditions when the leader UUV translates inx-y-z coordinate system. The results demonstrate that at the end of thescenario, the x-axis offset between the leader and the follower UUVs ismaintained at a steady-state distance of 8.25 m, the y-axis offset at0.04 m and the z-axis offset 0.03 m (FIG. 4G). The x-axis leader poseestimation during the time of flight is within reasonable accuracy andas a result, a smoother follower UUV trajectory is achieved. For z-axiscontrol, initially, the leader UUV's z-axis coordinate is estimated todecrease while it actually increases. The pose detection algorithm thencorrects its estimations and a more accurate control action isperformed. A similar situation is observed in y-control where it is seenthat the leader UUV is initially estimated to move in the reversedirection of its actual trajectory. Then, the detection algorithmcorrects its estimations and more accurate control feedback andtrajectory control are achieved. FIG. 4G shows the leader-follower casestudy in a dynamic-dynamic system with SMC for a 5×5 array. (a) x-axiscontrol (b) y-axis control and (c) z-axis control.

Discussion

Simulation results of pose detection algorithms and SMC demonstrate thatdynamic positioning of UUVs can be established with acceptable accuracy.The performance of the pose detection algorithms evaluated in thischapter (phase correlation and log polar transform, SAM, and imagemoment invariants) are compared based on their ability to provide fast,reliable and accurate pose estimates. The phase correlation andlog-polar transform algorithm yield accurate results for cases when themotion is only translation. For rotational motion detection, it is notsuitable as symmetric detected images of the external light sourceprevent accurate pose estimations. It should be stressed that roll angleis not detectable as there is a single light source with a Gaussianintensity profile. The results of SAM algorithm and its application onthe control of a leader-follower UUV application for a planar array havebeen shown. SAM algorithm can yield successful results. However, thecalibration procedure to provide pose outputs is extensive and a lengthylook-up table is needed to generate a pose output. Therefore, it may notrespond to the system requirement of a fast processing algorithm. Imagemoment invariants approach is chosen due to the following reasons:Simple linear models exist for pose estimation, which leads to fastercomputational speeds and makes it suitable for real-time applications.In addition, less calibration time is required as its dependence on thelook-up tables decreases. This results in less required computationaleffort in the implementation of the system. Furthermore, an accurateestimate of poses can be obtained with multiple concurrent non-zero poseerrors. More specifically, x-axis pose estimation can be obtained whenthe UUV undergoes y and z-axis translations and yaw rotations.Alternatively, a procedure can be developed to distinguish and quantifyconcurrent y-axis translations and yaw motions.

SMC and PID controllers are evaluated in conjunction with the feedbackobtained from the image moment invariants algorithm. The PID controllerresults in system overshoot. While overshoot can be tolerated in x-axiscontrol, overshoot in the y and z-axis directions or in yaw is notdesirable as it results in the UUV losing line of sight with theexternal light source. The SMC is selected as a first-order controllerwith a time-varying boundary layer and a saturation function (tominimize chatter). The controller is a SISO controller where thekinematic and dynamic cross-coupling terms are neglected in the UUVmodel. The uncertainties due to added mass and hydrodynamic forces arecompensated in the control system. SMC requires position and velocitystate information as sensor feedback. The position information can beobtained from pose detection algorithms. However, for the velocitysignals, the derivative of the pose information is taken. In order toavoid data fluctuations, the Kalman filter is implemented for bothposition and velocity signals. SMC simulation results suggest thatsatisfactory positioning results can be obtained. Overshoot which is animportant criterion in this study is not observed in SMC simulationresults. The study of the effect of external disturbances such ascurrent suggest that the control system can yield acceptable results(especially in x and y-axis directions) under a modest amount ofconstant current in x-axis (−0.03 m/s). However, the steady-state errorin yaw increases and requires compensation for reasonable accuracy.

The effect of the detector array size on the dynamic positioning isinvestigated by developing pose detection algorithms for a 21×21 and 5×5detector arrays. The simulation results conducted on a 5×5 array suggestthat the developed algorithms yield satisfactory control results, so asto deem the use of a more costly 21×21 array unnecessary. The SMCcontroller is shown to be robust against modeling uncertainties and tomodest amount of disturbances. However, when there are largerdisturbance forces (especially in the y-z plane and/or when there isnon-systematic background noise such as a sediment plume), the estimatedpose may be misinterpreted and the UUV could lose line of sight with theexternal light source.

Conclusions

UUV pose detection and control algorithms are evaluated to dynamicallyposition a UUV in scenarios of static-dynamic (regulation control withrespect to a fixed external light source) and dynamic-dynamic (trackingcontrol with respect to a moving UUV) control. Criteria for posedetection and control algorithms are: 1) processing time, 2) positionalaccuracy, and 3) dependence on environmental characteristics. Based on apreviously developed simulator that calculates the interaction of lightwith respect to water conditions and hardware characteristics, an imageis generated from the measurements on a hemispherical detector array.

Three pose detection algorithms (phase correlation and log-polartransform, SAM, and image moment invariants) are evaluated. It isdetermined from a series of simulations that the method of image momentinvariants requires a modest amount of calibration and is the mostsuitable in terms of processing speed and positional accuracy. Utilizingimage moments invariants algorithm, linear models are created todetermine and quantify the type of motion to differentiate and estimate3-DOF translational motion and pitch and yaw. Case studies are simulatedfor both static-dynamic and dynamic-dynamic systems.

PID control and SMC are also evaluated in terms of their responsecharacteristics. According to simulation results, it is concluded thatPID control is not suitable for this application as it causes a systemovershoot which, in turn, causes UUV loss of line of sight with thetarget light source. The SMC does not result in overshoot and isimplemented in a case study to demonstrate the UUV dynamic positioningperformance when it is given initial conditions such that there aremultiple concurrent non-zero pose errors. A dynamic-dynamic case studyalso suggests that a follower UUV can track a leader UUV under theseconditions with reasonable accuracy.

The effect of detector array size on the pose detection algorithms isevaluated to explore their capabilities to generate pose feedback forUUV dynamic positioning. For this purpose, the performance results of a21×21 and a 5×5 detector array are compared under independent DOFcontrol in all 5-DOF motion (x, y, z, yaw and pitch), demonstrating thata 5×5 detector array is sufficient to generate pose feedback from thesampled light field.

Experimental Pose Detection for UUV Control System Using an OpticalDetector Array

The capabilities of an optical detector array to determine the pose (x,y and z-axis) of a UUV based on optical feedback to be used in UUVdocking applications are demonstrated in this chapter. The opticaldetector array consists of a 5×5 photodiode array that samples theintersected light field emitted from a single light source and forms animage. After a set of calibrations for pose geometry, it is possible todevelop pose detection algorithms based on an image processing approach,specifically image moments invariants. Monte Carlo simulations wereconducted to determine the system performance under environmental andhardware uncertainties such as diffuse attenuation coefficient,temperature variations and electronic noise. A previously developedsimulator was used as a test bed to run the Monte Carlo simulations. Thesimulator takes the relative geometry between the light source and thedetector and environmental and hardware characteristics as inputs. Theperformance evaluation for Monte Carlo simulations was based on thegenerated pose outputs with respect to changing environmental andhardware parameters for a number of samples, i.e. N_(S)=2000.Experimental results of this study show that the pose estimations in x,y and z-axis are accurate within 0.3 m, 0.1 m and 0.1 m, respectively.Monte Carlo simulation results verify that the experimental results arewithin the confidence interval bounds (95%) and the pose uncertaintiesassociated with x, y and z-axis are 1.5 m, 1.3 m and 1 m.

Introduction

Unmanned Underwater Vehicles (UUVs) provide an operational platform forlong periods of deployment on the order of hours and in depths that aretoo dangerous for divers. However, the time of operation of thesesystems is limited based on the hardware available in the platform, suchas the power supply and data storage capacity. In order to extend theduration of the mission, the power supply needs to be replaced orrecharged and data should be transferred from the UUV's internal storageunit to an external storage unit in order to clear space for additionaldata collection. A common approach to extend the UUV operationsunderwater is the use of docking stations which enable the UUVs toconduct data transfer and recharge the batteries.

The two most common types of docking station architectures are: (1)funnel-docking station in which the UUV enter a tube for homing; and (2)pole-docking station, where the UUV connects to the station using a hookmechanism placed perpendicular to the seafloor. The structural design ofthe funnel docking station is similar to a cone (FIG. 5A). The funneldocking station is designed for a specific class of UUVs of the samelength with the same physical connections for power. The funnel dockingstation allows a small tolerance of misalignment as the UUV navigatesinto the cone. Due to the cone shaped design, the UUV's entrancetrajectory into the docking station is restricted. The data transfer inthe funnel type system can be conducted through wired communication orwireless Ethernet radios. The power transfer is accomplished by having acharge pin inserted from the docking mechanism. The data and powertransfer in these systems are reliable as there is a stable connectionbetween two platforms when the vehicle goes into the funnel type dockingstation. However, the design of each funnel-docking station is uniquefor a specific class of UUVs. The architecture of the pole-dockingstation offers an omnidirectional docking approach (FIG. 5B). The UUVlatches onto a vertical docking pole in order to dock. After the UUV issecurely latched, a circular carriage that moves along the pole forcesthe UUV to mate with the inductive links for data and power transfer.The pole docking station does not restrict the UUV's entrance trajectoryinto the docking station. However, the tolerance of UUV speed and itsdistance from the pole is limited. If the UUV is not rigidly attached tothe pole, power and data connections between the UUV and the dockingstation may not be successful. FIG. 5A shows a funnel type dockingstation and FIG. 5B shows pole docking mechanism architectures.

Recent studies have demonstrated the potential use of both acoustic andoptical communication for docking. In these systems, acousticcommunication is used in relatively longer ranges, 100 m, for navigatingtowards a docking station and video cameras are used in closer ranges,8-10 m, to guide the vehicle into the docking station.

This chapter presents pose detection to be used in UUV control systemusing the feedback from the optical detector array. This approach can beutilized in UUV navigation e.g. into a docking station. The detectionsystem developed is based on a static-dynamic system, i.e. one UUV isnavigating to a docking station that is fixed in space. A single beaconlight source at the docking station was used as a transmitting unit anda prototype detector array interface mounted on a dynamic UUV platformwas used as a detector unit. The input data generated from the detectorarray are signature images of the light field that were used tocalculate the relative pose between the UUV and the docking station andprovide feedback to guide the UUV to the docking station. The posedetection performance during the navigation of the UUV platform isevaluated for both the funnel-docking and pole-docking stations. Thesystem was designed based on the environmental characteristics ofPortsmouth Harbor, N.H. As a proof-of-concept, a scaled model wasexperimentally tested at wave and tow tank at Jere E. Chase OceanEngineering facilities.

The performance of the experimental platform, i.e. optical detectorarray, developed in this study was evaluated in terms of two criteria.The first criteria is accuracy of pose detections in three axes, x, yand z. This requirement is set based on a potential successful dockingoperation of a UUV especially for a funnel type docking station in whichthe entrance trajectory is restricted. The second criterion is theaccurate UUV velocity estimations. This criterion is important when theUUV approaches to docking station especially for a pole docking stationin which the UUV speed tolerance is limited rather than the entrancetrajectory.

Pose Detection and Hardware

The pose detection algorithms were developed based on the image momentinvariants approach which was previously described herein.

The hardware selection for this study is based on the results obtainedfrom prior research that included evaluation of different detector arraygeometries based on their capability to generate a unique pose feedbackto the UUV. The evaluation included curved and planar array designs withvarying number of photo-detector elements. It was concluded that a 5×5hemispherical curved array is sufficient to generate the desired posefeedback to the UUV. Analytical pose detection and control algorithmswere developed for both static-dynamic and dynamic-dynamic system, i.e.one leader UUV guides the follower UUV. During the development ofanalytical pose detection algorithms for both systems, the detectorarray was assumed to be mounted on the bow of the UUV. In thestatic-dynamic system, a single light source that acted as a guidingbeacon was mounted to the wall whereas in the dynamic-dynamic system,the light source was placed onto the crest of the leader UUV.

The detector module used in this research consists of a 5×5 photodiodearray (Thorlabs SM05PD1A), two Analog to Digital (A/D) boards, anon-board computer (OBC), power supply and reverse-bias circuit elements.The photodiodes placed on a hemispherical surface with an outer diameterof 0.25 m (FIG. 5C). The hemispherical surface with 25 holes of 0.0254 mdiameter was manufactured using a Rapid Prototyping Machine (DimensionSST 768) using ABS material. Each detector was placed in waterproofacrylic fixtures that were mounted onto the holes on the frame(photodetector facing outward) and aligned concentric to the hemispherecenter. The length of each acrylic fixture outside the hemisphere is0.064 m. Thus the effective radius of the detector array (radius of thehemisphere plus the length of the fixture) is 0.19 m. The photodiodeswere connected to a reverse-bias circuit that provided dynamic outputrange from 0 to 5 V. SubMiniature version A (SMA) cables were used toconnect the photodiode output to the reverse bias circuit. The lightintensities collected by the photodiodes were sampled using two A/Dboards on two different Arduino microcontroller platforms with 10-bitresolution (0-1023 bit range). The data sampled at the A/D boards weretransmitted serially to the OBC running a 1 GHz ARM Cortex processorwith Linux operating system. The OBC receives the collected lightintensity data from the photodiodes and sends it to a Linux based PC.The photodiode intensity readings were sampled at 5 Hz. The power supplyused in the reverse-bias circuit was provided by 5 V port on the Arduinoplatform. The reverse-bias circuit to increase the dynamic range of asingle photodiode consists of a 47Ω resistor, a 1 MΩ resistor and a 0.47μF capacitor. FIG. 5C shows photos of the optical detector array used inthe experiments. The photodiodes are facing different angles for anincreased field-of-view. They are placed on an ABS hemisphere surfacefor precise hole locations. The acrylic hemisphere is used forwaterproofing.

Methodology

The empirical measurements in the study were conducted in the wave andtow tank at the UNH's Jere E. Chase Ocean Engineering facilities. Themeasurements were based on scaled model on Portsmouth Harbor between NewHampshire and Maine. A prospective location for a docking station isconsidered at the entrance to the harbor near UNH's Judd Gregg MarineResearch Complex facilities in Fort Point, Newcastle, N.H. PortsmouthHarbor is a highly active port that includes a naval shipyard, fishingvessels, survey vessels and recreational vessels. As a result, theharbor is acoustically noisy and optical communication is the mostviable method to navigate a UUV to a docking station. There are severalfactors that affect the reliability of the system performance. Thesefactors can be listed as the diffuse attenuation coefficient, bathymetryand current information in the prospective implementation area.According to UNH Coastal Ocean Observation Center archive on Aug. 16,2005, the average diffuse attenuation coefficient value in PortsmouthHarbor area was 0.2 m⁻¹. The depths in the harbor range up to 20 m indepth at the center of the navigational channel with a current speedrange of 0.1-0.9 m/s at around 12 m depth at Fort Point.

The depth of the wave and tow tank is 2.44 m with relatively clear waterconditions (diffuse attenuation coefficient of 0.09 m⁻¹). The tank isoutfitted with a cable-driven tow carriage with actuation that extendsthrough the length of the tank that can move up to 2.0 m/s. A singlelight source was evaluated as potential guiding beacon for the dockingstation: 400 W metal halide light with ballast. This mock-up dockingstation is placed onto the wall of the wave and tow tank (FIG. 5D). Thedetector array was mounted on an aluminum frame on the wave and towtank. For pose calibration, the distance between the light source andthe detector array was measured and the x-axis offset was controlledwith the actuating mechanism. FIG. 5D shows the wave and tow tank at theUNH Ocean Engineering facilities. The tow tank is cable driven andcomputer controlled with 1 mm precision along x-axis. Left—detectorarray mounted on the dynamic platform on the Tow-Tank. Right—400 W lightbeacon used as a mock-up docking station.

Calibration Procedure

Two types of calibration procedures were conducted for this study: 1)Calibration for photodiodes: This step includes a) Output consistency ofthe photodiodes to be used in the optical detector array when they areexposed to the same light field conditions. b) Photodiode responses tothe potential temperature variations. c) The potential noise level andthe cross-talk of the system. 2) Calibration for pose estimation inwater in wave and tow tank: This step consists of the calibrationexperiments which in turn is to be used in the development of posedetection and control algorithms.

In the first calibration procedure, in order to check the consistency ofthe photodiode outputs, a single photodiode, i.e. photodiode under test,was mounted to a threaded cage plate (Thorlabs SM05). Photodiode andcage plate setup was stabilized at a distance of 0.18 m away from ahalogen light source (PL-900 Fiber-Lite). The output from the photodiodeunder test was connected to an Oscilloscope (Tektronix DPO 3054). Thedata was collected for 2 minutes and the average voltage was recorded.This procedure was repeated for all of the photodiodes on the array,i.e. 25 photodiodes (FIG. 5E). FIG. 5E shows the Photodiode CalibrationProcedure. 25 photodiodes were tested at a time in order to observetheir output voltage range under same conditions.

A separate calibration procedure was conducted to quantify SM05PD1Aresponse to the potential temperature changes in water. SM05PD1A placedin an acrylic waterproof housing was submerged into a digitallycontrolled refrigerated bath/circulator (NESLAB RTE-111). In thisprocedure, the water bath was used to change the surrounding watertemperature. The temperature of the water bath was changed from 20° C.to 70° C. at 10° C. increments. 20° C. is approximately the operatingtemperature at the wave and tow tank. A green laser at 532 nm wavelength(Z-Bolt SCUBA underwater dive laser) with the power output of 4 mWilluminated the photodiode. A k-type thermocouple was used to measurethe temperature of the photodiode and fixture. At each temperature, thesystem was allowed to come to thermal steady state before the voltageoutput of the photodiode was measured. The output of SM05PD1A wasconnected to an oscilloscope (National Instrument PXI 5142) and theresponses for varying temperatures were recorded (FIG. 5F). FIG. 5Fshows a diagram for temperature calibration (top) and an experimentalsetup for photodiode response to temperature changes (bottom). In thedisclosed experimental procedures a thermocouple was placed inside thewaterproof housing for temperature monitoring of the photodiode.

Photodiode response is also observed for any potential noise andelectronic cross-talk in the data acquisition system. In order to testthese effects, all of the photodiodes were mounted on the curveddetector array. In a dark environment, one photodiode was illuminated ata time with a light source using a black plastic tube between the lightsource and the detector. The remaining 24 photodiodes were exposed tothe ambient light. The response of each 25 photodiode was observed forany potential cross-talk that can occur during the transmission of thesignals with 3.3 m SMA cables. Potential noise sources in the hardwarewere explained in more detail.

The final calibration procedure, i.e. pose calibration procedure, wasconducted to detect and quantify the pose to be used as the feedbacksignal in the control system. The DOF of interest for the UUV motion atthe calibration stage are translations along x, y and z-axis. Thecalibration procedure was conducted using the optical detector array.Optical detector array was placed in the tow carriage in the wave andtow tank. The center of the light source was submerged 1 m deep in thewater column. The calibration procedure for x-axis was conducted usingthe computer controlled actuation mechanism in the wave and tow tank. Inx-axis, optical detector array was brought from 4.5 m to 8.5 m at 1 mincrements. For y and z-axes, aluminum 80/20 frames were used toquantify the amount of offset and bring the optical detector array tothe specified location. The calibration procedure for y-axis wasconducted from −0.6 m to 0.6 m at 0.3 m increments. For z-axis, thecalibration procedure was conducted starting at 1 m deep in the watercolumn to 1.8 m depth at 0.2 m increments. Thus, at each x-axisposition, 25 different images (beam patterns) were collected (125 imagesin total calibration scheme). The photodiode intensity data collectedduring the calibration procedure were analyzed offline in order todevelop algorithms that convert the light input into pose information.

Performance Evaluation

The performance of the system was evaluated in terms of two criteria. 1)Positioning accuracy of the UUV platform with respect to the lightsource for both in steady-state and dynamic cases. 2) Velocityestimation accuracy during the navigation.

For the first criterion, the UUV platform, which is the optical detectorarray mounted on the actuating tow tank, was set in the water column ata pre-determined offset away from the light source. Then, with aspecified acceleration and velocity in the tow tank controller, the UUVplatform was commanded to go to a final position. In the dynamic tests,the goal is to detect the location of the center of the light sourcewithin 0.5 m for x-axis and within 0.2 m for y and z-axis when the UUVplatform is both stationary and dynamic. This is a requirement for bothlocating the target, i.e. the docking station and for a successfuldocking operation. The tolerances for y and z-axis are tighter than theerror tolerance in x-axis as the UUV requires a certain accuracy in yand z-axis to enter into the docking station. Two sets of experimentswere conducted in order to verify the system performance. The firstexperiment consists of the case when the UUV platform and the lightsource are aligned in y and z-axis with an offset of 8.5 m in x-axis.The second experiment is conducted for the case when there are offsetsof 0.6 m in y-axis and 0.8 m in z-axis. These offsets were chosen as themaximum possible offset specified by the calibration range in order toevaluate the system performance in its most limiting conditions.

For the second criterion, during the motion of the UUV platform, it isalso important to estimate the velocity relative to the light source.Based on the velocity feedback obtained from the optical detector array,the UUV control system can control its speed during its navigation for asmoother entrance into the pole docking station. The velocity estimationperformance of the UUV platform was evaluated during the dynamicexperiments that were conducted for pose estimations. The system wasevaluated to yield reliable estimations if the velocity is within 0.1m/s of the reference velocity specified during the platform motion.

Stochastic Assessment of Pose Uncertainty

In addition to the empirical performance evaluation, an accuracyassessment of the pose detection algorithms was developed using MonteCarlo analysis. The accuracy assessment is used to predict theuncertainty in the final pose based on the optical feedback. All thefirst-order and second-order parameters contributing to the poseestimation were identified. These parameters included from environmentalcharacteristics such as diffuse attenuation coefficient and temperaturevariation in the water column to the detector array and processorhardware noise. By using a random distribution for a larger number ofsamples (e.g. N_(s)=2000) for these parameters, the total propagationuncertainty (TPU) can be obtained.

The parameters that contribute to the forming of the beam pattern on theoptical detector array mainly depend on the environmental conditions.More specifically, diffuse attenuation coefficient, i.e. a measure ofturbidity in the water, and the temperature variation in the mediumcontributing to environmental conditions. The scattering of light in thewater column is not taken into account in this study. The uncertaintiesrelating to diffuse attenuation coefficient and the water temperaturevariations were modeled as uncorrelated Gaussian random variables. Theuncertainty associated with the hardware characteristics was modeled asunipolar random values drawn from standard uniform distribution. Thesevariables were input to the developed simulator which generates an imageon the specified number of photodiodes based on 1) the geometry betweenthe light source and the detector 2) the environmental characteristicsof the medium such as turbidity and the temperature, and 3) hardwarecharacteristics such as electronic noise and light source intensityprofile and distribution, etc. Monte Carlo simulation scheme is shown inFIG. 5G. FIG. 5G is a Monte Carlo flow diagram for the pose statistics.Pre-determined model uncertainty parameters were integrated into thehardware and environment model to estimate the total uncertaintypropagation in the pose detection algorithms.

The standard deviation of the hardware noise was determined during thecalibration process and it was deduced that the standard deviation ofthe noise did not exceed 1% of the maximum photodiode intensity.Uncertainty parameters used in Monte Carlo simulations are given inTable 5.1

TABLE 5 Uncertainty parameters for Monte Carlo simulations ParameterStandard deviation Diffuse attenuation coefficient $0.015\frac{1}{m}$Temperature variation 3° C. Hardware noise 2 orders of magnitude lessthan the maximum intensity on a photodiode

Results

Calibration Results

The output signals from the photodiodes when the emitted light fieldintersects with the detector array are to be used in pose detection andcontrol algorithms. Therefore, the consistency of the photodiodereadings is vital and need to be characterized. In order to observe thephotodiode response, all of the photodiodes were set at the samedistance, 0.18 m, from the halogen light source (PL-900 Fiber-Lite). Thecalibration results showed that the photodiode readings were in therange of 291-300 mV with the mean value of 297.4 mV and the standarddeviation of 2.41 mV. This result shows that the photodiode measurementsat the same experimental conditions are reasonably close to each otherwithout much variation. Although the mean voltage variation forphotodiode measurements is not significant, it should be taken intoaccount in the pose detection algorithms.

In order to determine the temperature dependence of SM05PD1A in water,its response to the temperature variation was characterized for aspecific light source, i.e. Z-Bolt SCUBA underwater dive laser operatingat wavelength of 532 nm. For the calibration procedure, the temperaturewas varied from 20° C. to 70° C. at 10° C. increments. It was observedfrom the experiment that the voltage output from SM05PD1A decreases asthe temperature increases. During the experiments, it was noted that at70° C., the steam build-up in the water bath affected the amount oflight incident on the photodiode. Therefore, this data point wasevaluated as an outlier. By applying a linear fit line to the rest ofthe data points, temperature sensitivity was found to be 2 mV/° C. Theequation for the linear fit is

Vo(T)=−2*T+576  (5.1)

Here Vo(T) is the measured temperature in terms of voltage and T is thetemperature in ° C. Based on the surrounding temperature in theenvironment, the voltage reading can be adjusted and the effect of thevarying temperatures can be integrated in the pose detection and controlalgorithms.

After the calibrations for photodiode response range and photodioderesponse characteristics for varying temperatures were conducted, thesystem was investigated for noise and cross-talk that can be caused bythe signal transmission in the SMA cables or in their connection to thereverse-bias circuitry. It was observed from the data that when therewas no incident light on the other photodiodes, the noise level in thesystem contributed by the dark current, shot noise, background noise,quantification errors, cable transmission losses (including the 3.3 mSMA cables from the photodiodes to the reverse-bias circuitry and theserial communication losses) were in the range of 1 mV. Thus, it wasdeduced that the cable and circuit connection cross-talk in the systemis not significant.

Underwater calibration procedure was conducted in order to developalgorithms to determine the pose and the velocity during UUV platformnavigation. At each x-axis position, 25 images were sampled at differentlocations (5 different measurements at y-axis ranging from −0.6 m to 0.6m at 0.3 m increments and 5 different measurements at z-axis rangingfrom 0 m to 0.8 m at 0.2 m increments). The total number of images takenfor the pose estimation calibration is therefore 125. In order toobserve the variation in the photodiode readings, 200 measurements weretaken at each 125 locations. The measurements were averaged and thestandard deviations of the readings were recorded. It was found that forthe total x-axis calibration range, the variation in the photodiodereadings was around 1% of the maximum photodiode intensity.

Performance Evaluation Results

The success of the optical detector array depends on its pose andvelocity estimates when the UUV platform is both stationary andapproaching to the light source. There are several identifiers that canbe used to estimate the pose. For the x-axis pose estimation, the posedetection algorithm relies on the intensity of the middle photodiode.Based on the prior calibration experiments, an exponential fit wasapplied to the intensity readings on the middle photodiode taken atx=4.5 m to x=8.5 m at 1 m increments the center of the light source wasaligned with the middle photodiode. For the estimation of y and z-axisoffsets with respect to the light source, the algorithm relies on theimage moments invariants calculations. For each pose, the image momentsinvariants algorithm yields a 3×3 matrix which indicates symmetryinformation of the sampled light field. The pose detection algorithmdeveloped in this study utilizes matrix elements obtained from the 3×3matrix for y and z-axis and fits linear models to estimate the pose inthe corresponding axes. Because, in some examples, the x-axis estimationsolely relies on the middle photodiode intensity, the estimates can beinaccurate when the relative offset between the light source and themiddle photodiode increases. Thus, the pose estimation in x-axis iscorrected based on the y and z-axis pose estimates. The velocityestimation in x-axis is calculated by taking the derivative of theinitial x-axis pose estimates.

In order to validate the system performance for both stationary anddynamic cases, two types of experiments were conducted: 1) The center ofthe light source and the center of the UUV platform is aligned. 2) Thecenter of the light source and the UUV platform is at the maximum offset(i.e. at the maximum limits of the calibration scheme). In both of thesecases, the UUV platform is given an initial position of 8.5 m and afinal position of 4.5 m with given a velocity of 0.5 m/s andacceleration/deceleration values of 0.2 m/s².

Case 1: The Light Source and the UUV Platform is Aligned

In this case, the UUV platform is stationary at the beginning of theexperiment at x=8.5 m for 5.5 s. Then, the tow tank is commanded to goto 4.5 m (FIG. 5H). The offset of the UUV platform in y and z-axis arezero with respect to the light source.

FIG. 5H shows Experimental Case 1: Top left: Reference position, rawx-axis pose estimate, corrected x-axis pose estimates, and the movingaverage window result. Top right: Velocity reference, raw velocityestimates and moving average window of size 10 applied to the rawvelocity estimates. Bottom left: y-axis pose estimate and the appliedmoving average window of size 10 during the motion. Bottom right: z-axispose estimate and the applied moving average of size 10 during themotion.

The results for the first experimental case show that x-axis poseestimate is within 0.3 m accuracy at all times during both when theplatform is stationary and dynamic. The corrected and uncorrected poseestimates are very similar in this case as the estimations in y andz-axis poses are not off from the reference pose significantly. Becausethe velocity estimates are obtained by taking the derivative of thex-axis pose estimate, the velocity signal is prone to noise. However, byapplying a moving average window of size 10, it is observed that theestimated velocity trend follows the reference velocity within areasonable accuracy, less than 0.1 m/s. For y-axis pose, initially, whenthe platform is stationary at x=8.5 m, there is an estimation error ofaround 0.18 m. The error in the corrected x-axis estimation is higherwhen the platform is stationary due to the estimation errors in y-axisduring the same period. After the platform starts moving, the estimationerror reduces significantly to 0.04 m. The z-axis estimation is moreaccurate when the UUV platform is stationary with the maximum estimationerror of 0.05 m. When the platform starts moving, the error increases to0.08 m.

Case 2: The Light Source and the UUV Platform is at Maximum Offset

The second experimental case is identical with the first experimentalcase with the difference of y and z-axis offsets (FIG. 5I). In thisexperiment, y-axis offset is set to 0.6 m and z-axis offset is set to0.8 m lower than the light source (1.8 m below the water surface). TheUUV platform is commanded to go to final x-axis position of 4.5 m fromits initial x-axis position of 8.5 m away from the light source at 2.5s.

FIG. 5I shows Experimental Case 2: Top left: Reference position, rawx-axis pose estimate, corrected x-axis pose estimates, and the movingaverage window of size 10 result. Top right: Velocity reference, rawvelocity estimates and moving average window of size 10 applied to theraw velocity estimates. Bottom left: y-axis pose estimate and theapplied moving average window of size 10 during the motion. Bottomright: z-axis pose estimate and the applied moving average of size 10during the motion.

For the x-axis pose estimation, both corrected and uncorrectedestimations have offsets in the beginning of the motion when theplatform is stationary (uncorrected estimations have 0.25 m andcorrected estimations have 0.5 m offset). As the platform starts movingand finally comes to a stop at 4.5 m, the corrected pose estimationsyield more accurate results than the uncorrected estimations, with 0.05m error in corrected estimation and 0.8 m error in uncorrectedestimation. Thus, it was deduced that the corrected estimations forx-axis should be taken into account as the feedback signal in thecontrol algorithms. Similar to the first experimental case, the x-axisvelocity estimations are noisy due to the derivative operation. However,applying moving average reveals that the trend of the velocityestimation follows the reference velocity. Pose estimations in y-axisstarts with 0.2 m error when the platform is at steady-state conditionsat 8.5 m. When the platform moves towards the light source, the y-axispose estimation error reduces and stays within 0.1 m of the actualvalue. Although the pose estimation pattern for y-axis fluctuates, themoving average result shows that the accuracy of the estimations iswithin 0.1 m. Estimations for z-axis also start with 0.2 m error. Itapproaches to 0.1 m as the platform moves closer to the light source.The experimental results verify that the optical detector array yieldspose estimations within the specified error values for a successfuldocking operation.

Stochastic Model Results

In order to predict the performance of the pose detection algorithmsunder varying environmental and hardware conditions, such as inPortsmouth Harbor, N.H., a stochastic approach, such as Monte Carlosimulations, was taken. The goal of the Monte Carlo simulations is toevaluate the system performance when there is uncertainty in theenvironmental conditions (diffuse attenuation coefficient andtemperature variations) and hardware characteristics. The measurementsof these parameters are not always available or require extensiveexperimental work to obtain. The uncertainties in the environment andhardware affect the reliability of the light intensity measurements andthus the pose output for each axis. Therefore, Monte Carlo simulationswere conducted to evaluate the system performance under changingparameters.

Two sets of Monte Carlo simulations were conducted in this study. Thefirst simulation compares the experimental pose estimations to the modelgenerated pose estimations. The simulations were conducted to evaluatethe second experimental case. Here, under the same detector trajectory,i.e. starting from x=8.5 m to x=4.5 m, the model generated nominal poseswere calculated by changing the diffuse attenuation coefficient,temperature and hardware noise with sample size of, N_(S)=2000. Theexperimental pose, reference pose, upper and lower CI bounds (95%) forx, y and z-axes are shown in FIG. 5J, respectively. FIG. 5J shows anexperimental pose with the Monte Carlo generated CI bounds (95%). Thestandard deviation of the hardware noise is set to 1% intensity of thephotodiode with the maximum intensity.

The experimental pose estimations by the pose detection algorithm staywithin the Monte Carlo generated CI bounds (95%) for all axes. Themaximum observed uncertainty for x-axis is around 1.5 m while it is 1.3m and 1 m for y and z-axis respectively. The bounds for x-axisestimations do not exhibit much variation during the course oftrajectory. However, for y-axis, the uncertainty decreases until t=6.8sec. and then starts to increase as the detector approaches to its finalposition. For z-axis, the uncertainty has a decreasing trend after themotion starts and as the optical detector array gets closer to the lightsource it converges to a constant value.

The second Monte Carlo simulation scenario was conducted when aconceptual UUV navigates in xy-plane, with a predefined trajectory frompoint A to point B (FIG. 5K). The light source is assumed to be at x=0,y=0. Because the UUV navigates in xy-plane, it is likely that in thepose detection algorithms, the y-axis pose can be misinterpreted as yaw.Thus, the capability of the detection algorithm to distinguish andquantify y-axis and yaw was also evaluated in this scenario. The pitchangle was not taken into account in this study as the UUV is assumed tobe built stable in pitch. FIG. 5K shows Monte Carlo Simulation resultswith 95% CI bounds when the conceptual UUV navigates a zig-zagtrajectory in xy-plane. The standard deviation of hardware noise is setto 0.5% intensity of the photodiode with the maximum intensity.Top-left: Nominal x-axis pose estimations. Top-right: y-axis poseestimation. Middle-left: z-axis pose estimation. Middle-right: Yaw poseestimation. Bottom: UUV reference navigation in the xy-plane and thenominal estimation.

The error between the reference motion and the model generated nominalpose estimations for x and y-axes were within the specified tolerancelimits at all times during the navigation. The maximum observeduncertainties in the x, y and z-axis are 1.5 m, 0.8 m and 0.48 m,respectively. During y-axis translation, the CI bounds get narrower thanthe CI bounds when the UUV does not move in y-axis. However, when theUUV undergoes translation in y-axis, the motion can be interpreted asyaw. However, the quantified value of this cross-talk is less than 5°.In the yaw rotation plots, in three regions namely around t=4.5 s, t=39s and t=71 s the CI bounds for yaw, decrease significantly. At theselocations, the pose detection algorithms can distinguish y-axis motionfrom yaw rotation efficiently. The Monte Carlo simulation results forthe two cases suggest that the nominal pose estimations follow thereference trajectory, with a small yaw angle detection of less than 5°.

Discussion

The results confirm that the hardware and detection system of thedetector array can be applied in static-dynamic system applications,e.g. for both funnel-docking and pole-docking stations. Evaluation ofthe feedback signal from the experimental results showed that thecurrent design can estimate the translational pose up to 0.3 m inx-axis, 0.1 m in y and z-axis. These results suggest that the proposedsystem has a strong potential to be used in both types of UUV to dockingstation applications that is planned to be implemented in PortsmouthHarbor, N.H.

The experimental y and z-axis pose outputs from the optical detectorarray exhibit a zig-zag behavior at times, in addition, there areoffsets (bias) from the reference geometry. One potential contributor tothese phenomena is the vibration associated with the tow carriage duringthe motion. The vibration can cause the crabbing behavior and affect thephotodiode intensity results. The second potential contributor is thedrag force that the optical detector array experiences during the motionwhich causes differences between the calibration measurements and thetest case measurements. These two contributors should be measured tocompensate for the effects of these in the detection system. The thirdpotential contributor is the models that were developed to estimate thepose using the image moment invariants algorithm matrix elements. Themodel generated uncertainty was blended in the TPU at the end the MonteCarlo simulations. A more detailed study analyzing the effect of theuncertainty of these models into the overall system should be furtherinvestigated. The velocity of the UUV platform can be estimated within0.1 m/s of the actual velocity. However, because the velocity estimationrelies on the estimated pose, the signal is prone to noise. Thus, inorder to utilize the estimated velocity signal in the UUV control systemas a feedback signal, it should be filtered to have a smoother velocityreference. This is important in UUV control systems perspective.

Hardware improvements such as using a higher resolution A/D boards andoperation in clearer waters can improve the pose detection results. Fromthe architectural perspective, it is possible to design a larger orsmaller detector array that can fit to a variety of UUV classes (bothfor Autonomous Underwater Vehicles (AUVs) or Remotely Operated Vehicles(ROVs)) for a variety of pose detection applications. The major designfeatures of the hemispherical detector array are an effective radius of0.19 m and a 5×5 array of photodiodes. Increasing the diameter of thearray, using additional photodiode elements and using different detectorarray geometry other than a hemisphere can increase the pose resolution.The placement of the acrylic fixtures on the optical detector array alsoaffects the reliability of the pose estimations.

The range of calibrations was limited by two factors in this study. Thefirst factor is the combination of turbidity in the tank and thespecific light source used in this study, which impedes the use of fulltank dimensions as calibration range in x-axis. Only a single lightsource was used in this study, i.e. metal halide light source. Utilizinga laser, the range of detection could be extended greatly. Whenselecting the hardware components such as the photodiode and the lightsource, water clarity should be taken into account. The detector and thelight source pair that allows maximum light transmittance underwatershould be selected. The second factor that limits the calibration rangein y and z-axis is the physical dimensions of the wave and tow tank,i.e. width and length. This limited the measurements to ±0.6 m in y-axisand −0.8 m lower than the light source in z-axis.

In addition to the empirical measurements, TPU of the pose estimationwas calculated using the Monte Carlo approach. The simulation resultsshow that two of the most influential factors affecting the study is theturbidity of the water column and the hardware noise. These twocomponents affect the photodiode intensity readings significantly. As aresult, the uncertainty of the poses can increase drastically. Thetemperature variation results obtained from the calibrations was notfound to be a significant factor affecting the photodiode intensityreadings.

Image moments invariants method was utilized in this research in orderto develop pose detection algorithms to detect and quantify the relativemotion between the UUV and the light source. The pose detectionalgorithms were developed when there is no noise in the system. Thealgorithms can provide very accurate estimations when the noise isintroduced to the system (standard deviation is 0.5% and 1% of theintensity of the photodiode with the maximum intensity). This shows thatalgorithm has specific tolerance to noise. However, when excessive noiseis added into the system in the form of hardware noise (e.g. standarddeviation of the noise is more than 1% of the intensity value of thephotodiode with the maximum intensity), the pose detection algorithm canyield inaccurate results.

It is also important to distinguish between the y-axis and yaw motion inorder to inform the control system with an accurate pose feedback. Theresults suggest that the pose detection algorithms can providereasonable discrimination between y-axis translation and yaw when thereis only y-axis translation. However, at some cases, y-axis translationvalues could be interpreted as small angle yaw rotations (less than 5°in the nominal case). At these values, the photodiode intensity valuesfor small angle yaw rotation and respective y-axis values were veryclose to each other which makes the distinguishing process very complex,if not impossible. This small-angle effect was verified for the casewhen the motion is pure yaw, i.e. no y-axis translation is present byconducting a Monte Carlo simulation (FIG. 5L). FIG. 5L shows a closerlook at the cross-talk between yaw and y-axis cross-talk when there isonly yaw motion. At x-axis increments of 0.1 m, the conceptual UUV wasrotated from −15° to 15° at 3° increments. Monte Carlo simulation wasconducted with sample number of 2000 and hardware noise level of 0.5% ofthe photodiode with the maximum intensity.

It was observed from the results that between rotations of ±3 degrees,the yaw motion was interpreted as y-axis motion. These results from bothcases from the Monte Carlo simulations confirm that at small yaw angledetections, the control system should have an extra step to validatethat actual UUV motion is yaw. Utilizing more than a single light sourcecan decrease this cross-talk effect.

When calculating the TPU results for the environmental conditions inPortsmouth Harbor, the range of parametric uncertainties regarding theenvironmental conditions should be increased. The parameters that areexpected show greater variation in Portsmouth Harbor than in UNH OceanEngineering facilities are the diffuse attenuation coefficient due toturbidity. In addition, scattering which was not considered in the modelused in this study can be a significant factor that affects the observedbeam pattern. Although it was not found to be a significant factor, theseasonal water temperature variations in Portsmouth Harbor should alsobe taken into account in the pose detection algorithms. These effectsare expected to increase the overall uncertainty in the pose estimation.Another important parameter to be considered during implementation ofsuch a system in Portsmouth Harbor are disturbances such as current andwaves. In terms of mechanical design and construction, docking platformshould be constructed to withstand these disturbances. In addition, theUUV controller should be able to compensate for the potentialdisturbances.

The optical detector array design can also be used in applicationsbeyond UUV navigation and pose detection. The high sampling rate of theAvalanche Photo Diodes (APDs) can be used for short-range free-spaceoptical communication for data transfer. Also, detector array can beused as an in situ beam diagnostics tool for different light sources todetermine the light field and characterize the scattering environment ofthe water column. One of the electro-optical consideration were toreplace the APD detectors with small camera (e.g., commercially off theshelf cameras (COTS)). The benefits of COTS cameras are that they arecost-efficient, easily interfaced to the OBCs and they provideadditional spatial information which can improve the pose detectionresults. However, image extraction and processing time needed for posedetection might not be fast enough for real-time applications.

Conclusions

Optical communication for UUV navigation and docking provides anaccurate and cost-effective positioning system that can yield relativepose information of a UUV. Especially, in environments that contain highlevels of acoustic noise. The goal of this study was to evaluate the useof an optical detector array unit for pose detection of the UUVs withrespect to a docking station using a light source as a guiding beam. Ascale model was used to simulate the environmental conditions ofPortsmouth Harbor, N.H. at the UNH Ocean Engineering Tank facility.Empirical measurements evaluated the performance of the unit withrespect to reference positions and velocities and a Monte Carlo analysisestimated the total propagation uncertainty in the system. Theexperimental results in UNH Ocean Engineering facilities showed thatpose detection in translational axes are within the required posespecifications. The pose estimations were within 0.3 m, 0.1 m and 0.1 mfor x, y and z-axis, respectively.

Monte Carlo analysis indicates that the stochastic assessment of theuncertainty in the system was consistent with the experimental results.The results demonstrate that at extreme conditions, i.e. highest amountof x-axis pose estimation uncertainty (95%) was on the order of 1.5 m,y-axis pose estimation uncertainty was 1.3 m and z-axis pose uncertaintywas on the order of 1 m. The analysis of the variables separately usingMonte Carlo approach showed that the most effective parameter on theuncertainty of the pose estimation is the diffuse attenuationcoefficient. Large variation in the hardware noise also had significantimplications on the overall pose estimation uncertainty. The algorithmalso demonstrated reasonable discrimination power between y-axistranslation and yaw rotation. The faulty detections were small angle yawrotation less than 5°. In the field implementation, i.e. in PortsmouthHarbor, the pose detection uncertainty in 4-DOFs, x, y, z and yaw, areexpected to increase due to higher turbidity and environmentaldisturbances such as waves, currents.

Discussion

In order to fully utilize the optical feedback in underwater,characterization of the underwater environment is essential. Experimentsconducted in wave and tow tank show that the effective range between thelight source and the light detector is dependent on the diffuseattenuation coefficient as it affects the light intensity during itstravel in underwater. In addition, the size of the beam pattern emittedfrom the light source has also a role in determining the usable portionof the light perpendicular to the optical axis. The approximatedimensions of the optical detector array can be determined based on thelocation where the light intensity starts to decrease to a certain valuesuch as FWHM.

Because a single light source with a Gaussian light intensity profile isassumed in this study, it is not possible to detect roll changes. Inaddition, in terms of practical application, most UUVs are built stablein roll and pitch, i.e. there is a specific limit for rotations in theseDOFs. Therefore, roll detection is not taken into account in this study.The use of multiple light sources or light sources with differentintensity patterns other than Gaussian intensity profile has thepotential to add roll detection capability to the detection system. Themaximum effective range between the light source and the light detectorwas kept at 8.5 m in this study. Utilizing lasers, e.g. a green laser at532 nm wavelength, can increase the range significantly.

The detector array can detect translational changes of 0.2 m androtational changes (pitch and yaw) of 10°. A larger array size is neededto detect smaller changes. However, a larger array size may not besuitable for all UUV types. While some UUVs such as a larger work-classROVs can handle array sizes with bigger dimensions, for the observationclass ROVs or an AUV, the mechanical stability can degrade by changinglocations of center of gravity and center of buoyancy. Another factoraffecting the detection capability is the light source characteristics.A narrow beam light source can provide better resolution detections bothin translation and rotation. Light sources with different intensityprofiles other than Gaussian can also change the detection capabilities.

The pose feedback is obtained by converting the beam pattern sampled onthe optical detector array into usable pose information through imageprocessing algorithms. Among these algorithms used in this study arephase correlation and log-polar transform, Spectral Angle Mapper (SAM),and image moments invariants. Phase correlation and log polar transformalgorithm is able to yield reliable translation estimations. However, itis not good for rotational estimations due to the symmetry of thesampled images. SAM algorithm proved to be useful for verifying the posedetection capability of the array. However, in terms of controlalgorithm development perspective, the SAM algorithm requires extensivecalibration procedure. It also increases the number of characteristicimage parameters to be used in a look-up table for distinction betweenthe motion types. This can increase the processing time and thereforeresult in a slower UUV thruster response which is undesirable. Imagemoments invariants algorithm yields reliable estimations for bothtranslation and rotation. In addition, the processing time to obtain thepose estimates is faster due to the linear models obtained from thecalibration procedure. In addition, the calibration procedure is not aslengthy as the SAM algorithm. Therefore, image moments invariantsapproach is chosen to extract the pose from a sampled image.

Two types of controllers were evaluated in this study,Proportional-Integral-Derivative (PID) and Sliding Mode Controller(SMC). The response characteristics such as overshoot were identified inconjunction with the feedback obtained from the image moments invariantsalgorithm. The analytical results for a static-dynamic system suggestthat PID controller exhibit excessive overshoot which is not tolerablein this application. When overshoot occurs during alignment with thelight source, the UUV goes out-of-sight and loses communication with theguiding beacon. The overshoot can be tolerated in x-axis but for y andz-axis control, this should be avoided. Control simulations with SMCsuggest that satisfactory performance could be obtained in bothstatic-dynamic and dynamic-dynamic system. However, a time-varyingboundary layer must be implemented with SMC in order to avoid chatter.In addition, SMC requires a full-state feedback, i.e. it should receivemeasurements for both position and velocity. The developed opticaldetector array can yield only pose information. The time-derivative ofthe pose measurements must be taken and filtered in order to have asmooth reference for the controller. Overshoot was not observed in SMCwhich makes it a more viable choice over PID as a controller. For thecontrollers that require extensive knowledge about the vehicle model andparameters, system identification of the vehicle should be conductedthrough experimental work. Other type of controllers such as H_(∞) andadaptive controllers should also be investigated for their practicalityin UUV control with optical feedback applications. Additional sensorsthat measure the disturbances and implementation of observers cancontribute for a better control system. It was also found thatsatisfactory control performance can be obtained by using a minimal 5×5array and thus use of larger number of arrays is not necessary. This isan important finding as it eliminates the costs associated with extrahardware such as large number of photodiodes, which in turn requirelarge number of ADC boards, extra OBCs and reverse-bias circuitry. Thisalso reduces the logistical complexity for the hardware implementation.

The accuracy of the detections was tested for two types of potentialdocking applications, i.e. funnel docking and pole docking. In funneldocking applications, the UUV entrance trajectory is more important thanits speed. Therefore, for this application pose estimation accuracy iscritical. For the pole docking applications, the UUV velocity is moreessential. Thus, the velocity estimations have more importance in thiscase. The pose detection algorithms were developed based on sampled beampatterns for a total number of 125 positions. Based on this set ofcalibrations, to estimate the pose, linear models were developedutilizing the image moments invariants matrix identities. Theperformance of the optical detector array was evaluated for bothaccuracy of pose estimations and velocity estimations. The experimentalevaluation of the pose detection of optical detector array demonstratedthat the developed system can determine the pose with 0.3 m accuracy inx-axis, and 0.1 m in both y and z-axis. Velocity estimates were within0.1 m/s of the actual tow velocity.

There are several factors that contribute to the accuracy of the posedetections using optical feedback. The physical affects that contributeto the errors in the measurements can be listed as the vibration of thetowing mechanism on the wave and tow tank and the drag force exerted onthe optical detector array. These affects cause the array to displaceand as a result the real-time intensity measurements differsignificantly from the offline calibration measurements at the samelocations. The setup used in the experiments consists of a single 80/20aluminum frame submerged into the water. One potential solution tominimize these effects on the pose estimations is to use a more stablesetup for calibrations. In addition, the placement of the acrylicfixtures on the hemispherical surface also is a source of systematicerror for the pose estimations. Another potential contribution to thepose errors is the linear models obtained from the calibration procedurethrough image moments invariants algorithms. Because the calibrationmodels are linear, the pose estimates are not equally accurate at eachposition.

The uncertainty analysis through Monte Carlo simulations revealed thatmost important factors affecting the pose estimation accuracy is theturbidity in the water and the noise which could be due to bothenvironmental effects (background noise) and hardware noise. It was alsoobserved that the temperature variation did not have a significanteffect on the estimations. Monte Carlo simulations were run for 2000samples with varying diffuse attenuation coefficient, hardware noise andtemperature values. The samples for these parameters were generated withrespect to normal distributions with an assumed known mean and standarddeviations. The pose detection algorithms were developed for zerohardware noise in the system and the algorithms proved to be robust tonoise with standard deviation more than 1% of the intensity value of thephotodiode with the maximum intensity. However, as expected, increasingthe amount of hardware noise resulted in additional uncertainty in thesystem. If the background noise in the environment is excessive orturbidity is high, the algorithms may not yield accurate estimations.However, this is an inherent issue with this application.

It is also of importance to distinguish motion that occurs in the sameplane. For example, y-axis translation and yaw act on the same plane andtherefore it becomes complex to discriminate between these two differentmotion types. After a calibration procedure, it was found that for themost part of the range under test, i.e. ±15°, y-axis translation and yawcould be distinguished. The failure to discriminate between y-axistranslation and yaw occurs when yaw is less than 5°. Around this angle,the corresponding intensity values on the optical detector array werealmost identical with the translation counterpart. This makes it acomplex process to distinguish between these two motion types,especially in turbid and/or noisy environments.

In the real-world implementation of such a system such as in PortsmouthHarbor, the pose estimation uncertainties are expected to increase dueto increased diffuse attenuation coefficient as a result of increasedturbidity in the harbor. Scattering which was not considered in thisstudy can also be a significant factor affecting the beam patternincident on the optical detector array. In addition, background noisewhich can be evident in the form of sediment plumes is expected toincrease the pose uncertainty. By evaluating the data from NOAA Charts,the environmental disturbances such as waves and currents are alsoexpected to increase in Portsmouth Harbor. The docking station should beconstructed to withstand these environmental forces. UUV control systemalso should compensate for these disturbances while navigating into thedocking station.

Conclusions

This study has shown the proof of concept of an optical detection systemthat can yield satisfactory pose estimation results. Pose detectioncapabilities and limitations have been demonstrated for static-dynamicand dynamic-dynamic systems through simulation and experiments. Thelimitations of the system in real-world conditions have been identified.

In order to design such a system, as the first step, underwaterenvironment was characterized in order to evaluate the feasibility of adetection and control system using optical feedback. From theexperiments, it was found that in 500-550 nm wavelength band, the lighttransmission is at maximum. FHWM radius of the beam expands from 0.3 mto 0.4 m from 4.5 m to 8.5 m distance. These findings yield a foundationfor hardware selection and dimensions of the proposed optical detectorarray design.

Two types of optical array geometries were evaluated for pose detectionin underwater, i.e. planar and curved arrays. The selection criteria forthe optical array design include the following: 1) the array shoulddistinguish changes in position and orientation based on a single lightsource. The array is expected to distinguish motion in 5-DOF, i.e.translations in x, y and z-axis as well as pitch and yaw. Due to thebeam symmetry emitted from a single light source, it is not possible todetect roll. 2) The array should have minimum number of opticaldetectors that can yield pose information, reduce processing time andreduce the costs. In order to evaluate the performance criteria, anumerically based simulator that takes the relative geometry between thelight source and the detector as inputs, the water turbidity, hardwareand background noise was developed. SAM algorithm evaluated the changesin position and orientation. Using a 21×21 array, it was shown that thetranslational changes of 0.2 m and rotational shifts of 10° can bedetected. The curved array was also shown to be more sensitive to thechanges in rotation whereas the two arrays performed similar for thetranslational shifts. Further simulations showed that a minimum of 5×5array is required to distinguish changes in 5-DOF. Experimentalmeasurements verified the accuracy of the simulator generated images.

The goal of this research is to use the optical feedback generated fromthe optical detector array in control applications. Pose detections andcontrol algorithms were developed in order to dynamically position theUUV in static-dynamic system, i.e. a fixed light source as a guidingbeacon and a UUV, and dynamic-dynamic system, i.e. moving light sourcemounted on the crest of a leader UUV and a follower UUV follows itspath. The algorithms were evaluated based on processing time, positionalaccuracy and dependence on the environmental characteristics. Ahemispherical array of sizes 5×5 and 21×21 was used to develop thealgorithms. Evaluation of image processing techniques such as log-polartransform and phase correlation, SAM and image moments invariantsdemonstrated that image moments invariants method is the most suitablealgorithm in terms of processing time and positional accuracy. Thedependence on the noise is an inherent issue for this application andunder excessive noise the accuracy of the results degrades for all ofthe algorithms. Evaluation of PID and SMC for static-dynamic anddynamic-dynamic cases demonstrated that PID is not suitable for thisapplication as it creates an overshoot during dynamic positioning,causing the UUV to lose line of sight with the light source. SMC on theother hand did not yield excessive oscillations and showed satisfactoryperformance for both static-dynamic and dynamic-dynamic applications.Analysis of the effect of detector number on the array revealed that a5×5 detector array is sufficient to generate pose feedback to be used inUUV control applications.

After analytical results that compare the effect of detector size,geometry, image processing algorithms and control algorithms on theaccuracy of pose estimations, the optical detector array wasexperimentally built. The final prototype consists of a hemispherical5×5 array with an effective radius of 0.19 m. The underwatercalibrations were conducted for 125 different positions in x, y andz-axis. Pose detection algorithms were developed based on thecalibration results using image moments invariants algorithm. Underwaterexperiments conducted in wave and tow tank showed that pose detectionaccuracy was within 0.3 m in x-axis and it was within 0.1 m in y andz-axis. The velocity estimations were also within 0.1 m/s. Thestochastic assessment of the pose estimations was done using Monte Carlosimulations. The Monte Carlo simulation results show that theexperimental pose estimations are within model generated CI bounds(95%). It was also demonstrated there is reasonable discrimination powerbetween y-axis translation and yaw rotation. The faulty detections weresmall angle yaw rotation less than 5°. The results also demonstrate thatat the extreme calibration conditions, the x-axis pose estimationuncertainty (95%) was on the order of 1.5 m, y-axis pose estimationuncertainty was 1.3 m and z-axis pose uncertainty was on the order of 1m. In the field implementation, i.e. in Portsmouth Harbor, the posedetection uncertainty in 4-DOFs, x, y, z and yaw, are expected toincrease due to higher turbidity and environmental disturbances such aswaves, currents.

Photodiode Data Collection Procedure Using Beagleboard-XM and TwoArduinos

This section explains the data collection procedure using the onboardcomputer Beagleboard-XM (BB-XM) and two Arduinos used as A/D converters.Any photodiodes can be used in this type of setup and here HamamatsuS1133 will be used. In the actual experiments Thorlabs SM05PD1A will beused.

The data collection procedure can be completed as follows. Photodiodesintersect the light and converts that into current. A voltage readingacross the terminals is done by two separate Arduino A/D input pins of10-bits, i.e. 0-1023 bits. Currently, there are 9 photodiodes so therewill be 5 photodiodes on one Arduino and 4 on the other. Arduinos areconnected serially to the BB-XM via the USB ports. BB-XM is connected toa PC through a RS-232 (on BB-XM) to USB cable (on PC side) (FIG. 6A).Python serial libraries are used for communication. In addition, all thecode to collect the data is written in Python programming language. Thissection is divided into two sections, i.e. A) Arduino and B) BB-XM. FIG.6A shows a photodiode-PC communication general diagram.

A) Arduino

-   -   1) Upload the Arduino sketches (The Arduino sketch for this        application is in the folder hmtsu. The program's name is        hmtsu.ino) to the Arduino Boards on a PC. Make sure that the        program is uploaded separately to the two boards and modify the        program accordingly. For example, there are 25 photodiodes and        so Arduinos will get 13 and 12 PD readings accordingly. Change        the number of PDs specified in the sketch accordingly (FIG. 6B).    -   2) Disconnect the Arduino from the PC.        FIG. 6B shows an example Arduino sketch. The example shown will        be for one Arduino.        For the other Arduino change the 13 to 12 and other variables        accordingly

B) BB-XM

In order to access BB-XM, we use the linux (Ubuntu) operating system.

-   -   1) Powering the BB-XM. We use a RS-232 to USB cable here. RS-232        cable is connected to the BB-XM and the USB end connects to the        PC. Connect the Arduino USB cables to the two of BB-XM USB        ports. Make sure the SD card is in BB-XM. BB-XM will be powered        from PC in experiments (We will connect a battery in later        stages). After making sure everything is connected, connect the        barrel end of the 5V barrel jack to USB power cable to the BB-XM        and the USB end to the PC (FIG. 6C).    -   2) Connect to the BB-XM from a PC through minicom. Open a        terminal in Ubuntu by Ctrl+Alt+T. In the terminal window type:        minicom. It should display Initializing Modem on the screen. (If        it does not connect, the port name might not be correct. In this        case find the correct port name and type minicom −D/dev/ttyUSBX.        Note: ttyUSBX is the device name. You can find the device name        by typing cd/dev and then type 1s). (FIG. 6D-E). FIG. 6D shows a        Minicom login screen. FIG. 6E shows a serial to USB port check        on PC.    -   3) After accessing BB-XM through minicom, arm login will not        accept what you type as password initially. At the second        attempt, type armlogin: firat, password: 1985fir. Remember that        you will not be able to see what you type in the password prompt        area (FIG. 6F). FIG. 6F shows an arm login and password screen.    -   4) Similarly, you can check the Arduino device names on BB-XM by        typing cd/dev and then typing 1s in the minicom terminal. The        Arduino device names are typically ttyACM0 and ttyACM1. These        port names are very important while establishing the serial        connection between BB-XM and the PC. (FIG. A.7)    -   5) For serial communication protocol and for data collection        code, we use Python programming language in both platforms, PC        and BB-XM.    -   6) To access the data collection program in BB-XM, type cd rov.        (Note: If you are in /dev folder, then you need to go back to        home by typing cd ˜) All the programs are in the rov folder.        Then type 1s to see the programs. The program to collect data        from Arduinos to the BB-XM is readPD.py. FIG. 6G shows a Arduino        Device names verified in the BB-XM.    -   7) In order to access the contents of the program readPD.py,        type vim readPD.py (FIG. 6H). vim is the name of the text        editor. Make sure the device names match with the connected        device names. IMPORTANT: If you need to edit the file, you need        to go into insert mode. Do this by hitting I button on keyboard.        When it is in insert mode, you will see—INSERT—at the bottom of        the program. If you are not in the insert mode, you cannot        change the contents. When you are in the INSERT mode:        -   To save and exit type “:wq”        -   To exit without saving, type: “:q!”        -   To exit only if you have not changed anything, type: “:q”        -   FIG. 6H shows a program that reads data from two Arduinos            and passes it to the pc. (readPD.py).    -   8) The data collection and observation will be on the PC side.        Open a new terminal on PC (Ctrl+Alt+T). Then type cd        Desktop/test. The name of the program for data collection is        getPD.py. You can also access the program by clicking the test        folder in Desktop and double clicking on the getPD.py program.        (FIG. 6I). This program reads the serial output coming from the        BB-XM and saves it into a text file. FIG. 6I shows the program        that reads the serial output of BB-XM and saves it to a file        (getPD.py).    -   9) To collect data, go to the minicom terminal, go to the rov        folder and type python readPD.py. You will see the series of        numbers printed out on the minicom terminal.    -   10) Now go to the PC terminal to the test folder on Desktop.        Type python getPD.py. You will see an error saying that the        device reports open but no serial data. Go to the minicom        terminal and close it. If minicom screen is open, there will be        no data collection.    -   11) Go back to the PC terminal and once again type python        getPD.py. The data collection will start.        IMPORTANT: You can kill the data collection by pressing CTRL+C        on the PC terminal. For data collection at a different geometry,        change the filename to your geometry for later analysis, and        repeat step 11. Everything will be changed on the PC screen from        now on. Minicom will keep running.        Beam Patterns from Experimental Data

1) 4.5 m

-   -   FIG. 7A shows beam pattern images at x=4.5 m

2) 5.5 m

-   -   FIG. 7B shows beam pattern images at x=5.5 m

3) 6.5 m

-   -   FIG. 7C shows beam pattern images at x=6.5 m

4) 7.5 m

-   -   FIG. 7D shows beam pattern images at x=7.5 m

5) 8.5 m

-   -   FIG. 7E shows a beam pattern images at x=7.5 m

Programs for Experimental Data Collection and Analysis

This section includes the programs that were written for photodiode datacollection. The photodiode readings are first sampled in the Arduino A/Dsketch, i.e. Arduino environment program. Then, the readings from twoArduinos are received by the BB-XM in two different USB ports. Theprogram that resides in the PC, i.e. readPD.py, saves the readings in atext file. Then a separate MATLAB file was written to read the contentsof the text file.

1) Program to Collect Photodiode Data to Arduino (hmtsu.ino)

int pins[13] = {0, 1, 2, 3, 4,5,6,7,8,9,10,11,12}; int sv[13]; voidsetup( ) {  // initialize serial communication at 9600 bits per second: Serial.begin(9600);  Serial.println(“Serial comm starts!”);  } // theloop routine runs over and over again forever: void loop( ) {  charcommand;  if( Serial.available( ) )  {   command = Serial.read( );   if( command == ‘!’ )   {    // read the analog in value:    for( inti = 0; i < 13; i++ )    {      sv[i] = analogRead( pins[i] );    }   //write the analog reads from the    Serial.print(“sensor = ” );   for( int i = 0; i < 13; i++ )    {     Serial.print(sv[i]);    Serial.print(“, ”);    }     Serial.print( “\n” );    // wait 2milliseconds before the next loop    // for the analog-to-digitalconverter to settle    // after the last reading:    delay(2);   }  } }2) Program to Get the Photodiode Readings from Arduino to BB-XM(readPD.py)

# code to read serial out of arduino import serial import re import timeser = serial.Serial(‘/dev/ttyACM0’, baudrate = 9600) ser2 =serial.Serial(‘/dev/ttyACM1’, baudrate = 9600) pc =serial.Serial(‘/dev/ttyO2’, baudrate = 115200 ) while True:  ser.write(“!”)   data1=ser.readline( )   ser2.write(“!”)  data2=ser2.readline( )   time.sleep(0.1)   pc.write(data1+data2)3) Program to Get the Photodiode Readings from BB-XM to PC and Save as a.txt File (getPD.py)

# program to get the serial data from BB-XM # use this file to collectdata import serial import time import re # make sure the connectionnames are correctsercom=serial.Serial(port=‘/dev/ttyUSB0’,baudrate=115200,parity=serial.PARITY_NONE,stopbits=serial.STOPBITS_ONE,bytesize= serial.EIGHTBITS)sercom.open( ) sercom.isOpen( ) #make sure to change the filename tocorrespond to the test coordinate f=open(‘single arduino.txt’,‘w’) whileTrue: #   bytesToRead = sercom.inWaiting( )    x=sercom.readline( )   f.write(time.ctime( )+‘,’+x)    f.flush( ) #   print time.ctime()+‘,’+x f.close( )4) Program that Extracts the Intensity Readings from the Text File andProcesses

% Data processing program that reads in the txt files and parses thevalues % into photodiode values clear;clc; close all %% 1) Data Parsingcount=1; % fileID=fopen(‘x4.5_y0_z1_upright.txt’,‘r’); %1 xi=4.5;zi=1.2; yi=0.3; % for xi=4.5:8.5    for zi=1:0.2:1.8     foryi=−0.6:0.3:0.6 fileID=fopen(sprintf(‘x%2g_y%1g_z%1g.txt’,xi,yi,zi));N=5; data=fread(fileID, [1 inf], ‘*char’); % Number of detectors onArduino 1 and 2 PD1=13; PD2=12; Psum=PD1+PD2+2; % This property is usedfor data parsing % find 2014 in the file.. Change this to 2015 in thenext year cycles = strfind(data, ‘2015’); fclose(fileID); %Day_ind=strfind(data,Day); comaind=strfind(data,‘,’); %Day_ind=Day_ind(2:end)’ k=2; if comaind(14)−comaind(13)<=5 fori=1:length(cycles)/2 Ard1(i,:)=str2num(data(cycles(2*i−1)+5 :comaind(Psum*i−PD1))); Ard2(i,:)=str2num(data(cycles(2*i)+5:comaind(Psum*i))); k=k+1; end else PD1=12;PD2=13; fori=1:length(cycles)/2 Ard2(i,:)=str2num(data(cycles(2*i−1)+5 :comaind(Psum*i−PD1−2))); Ard1(i,:)=str2num(data(cycles(2*i)+5:comaind(Psum*i))); k=k+1; end end PD=[Ard1 Ard2]; %% Plot of the averagePDsum=mean(PD); stdev(count,:)=std(PD); for i=1:5 B(i,1:5)=PDsum(1,5*i−4:5*i ) ; end subplot(5,5,count) imagesc(B)caxis([0 1023]) %% 2) Moment Calculation [row,col]=find(B==max(max(B)));row=row(1); col=max(col); % Sometimes multiple values are returned. Willpick max row; %% Sub-pixel accuracy algorithm % First derivatives dIdx=(B(row,col+1)−B(row,col−1))/2; dIdy= (B(row+1,col)−B(row−1,col))/2; %Second derivatives d2Idx2=B(row,col+1)−2*B(row,col)+B(row,col−1);d2Idy2=B(row+1,col)−2*B(row,col)+B(row−1,col); % Second partialderivatives d2Idxdy=0.25*(B(row+1,col+1)+B(row−1,col−1)−B(row−1,col+1)−B(row+1,col−1)); H=[d2Idx2 d2Idxdy; d2Idxdy d2Idy2]; D=[dIdx;dIdy];S=−inv(H)*D; col_sub(count)=col+S(1); % Calculate the sub-pixel accuracyrow_sub(count)=row+S(2); title(sprintf(‘x=%1gm y=%1gm z=%1gm’,xi,yi,zi)) xf=1:N; yf=1:N; [Ypos,Zpos]=meshgrid(xf,yf);Ysort=reshape(Ypos,N*N,1); Zsort=reshape(Zpos,N*N,1); i=0;li=1; fori=0:2   for j=0:2     k=1;     for k=1:N*N    Mom(k,li)=((Ysort(k)−col_sub(count)){circumflex over ( )}i) *((Zsort(k)− row_sub(count)){circumflex over ( )}j) *PDsum(k);    Mom_norm(k,li)=Mom(k,li)/sum(PDsum);      end     li=li+1;   end endMnt=sum(Mom_norm);Mmnt(3*count−2:3*count,:)=transpose(reshape(Mnt,3,3)); count=count+1;    end   end % end sprintf(‘row values are between%1g−%1g’,min(row_sub), max(row_sub)) sprintf(‘col values are between%1g−%1g’,min(col_sub),max(col_sub))

1. An optical detector comprising: a plurality of light sensors; atleast one controller coupled to the plurality of light sensors andconfigured to: receive a plurality of signals acquired from a lightsource by the plurality of light sensors; determine a first pose of theoptical detector relative to the light source based on the plurality ofsignals; determine a second pose of the optical detector relative to thelight source based on the plurality of signals; calculate a differencebetween the second pose and a target pose; determine a velocity of theoptical detector relative to the light source based on the difference;determine at least one command based on the difference and the velocity;and transmit the at least one command to adjust a trajectory of theoptical detector.
 2. The optical detector of claim 1, wherein theplurality of signals are acquired from light having a spectral rangebetween 500-550 nanometers.
 3. The optical detector of claim 1, furthercomprising a memory coupled to the at least one controller andconfigured to store data defining the target pose to be within 8.5meters of the light source.
 4. The optical detector of claim 1, furthercomprising an array housing the plurality of light sensors and having alength of 0.6 meters.